Fundamental mistuning model for determining system properties and predicting vibratory response of bladed disks

ABSTRACT

A reduced order model called the Fundamental Mistuning Model (FMM) accurately predicts vibratory response of a bladed disk system. The FMM software may describe the normal modes and natural frequencies of a mistuned bladed disk using only its tuned system frequencies and the frequency mistuning of each blade/disk sector (i.e., the sector frequencies). The FMM system identification methods—basic and advanced FMM ID methods—use the normal (i.e., mistuned) modes and natural frequencies of the mistuned bladed disk to determine sector frequencies as well as tuned system frequencies. FMM may predict how much the bladed disk will vibrate under the operating (rotating) conditions. Field calibration and testing of the blades may be performed using traveling wave analysis and FMM ID methods. The FMM model can be generated completely from experimental data. Because of FMM&#39;s simplicity, no special interfaces are required for FMM to be compatible with a finite element model. Because of the rules governing abstracts, this abstract should not be used to construe the claims.

REFERENCE TO RELATED APPLICATION

[0001] The disclosure in the present application claims prioritybenefits of the earlier filed U.S. provisional patent application serialNo. 60/474,083, titled “Fundamental Mistuning Model for DeterminingSystem Properties and Predicting Vibratory Response of Bladed Disks,”filed on May 29, 2003, the entirety of which is hereby incorporated byreference.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH

[0002] The invention in the present application was made under a grantfrom the United States Air Force Research Laboratory, Contract No.F33615-01-C-2186. The United States federal government may have certainrights in the invention.

BACKGROUND

[0003] The present disclosure generally relates to identification ofmistuning in rotating, bladed structures, and, more particularly, to thedevelopment and use of reduced order models as an aid to theidentification of mistuning.

[0004] It is noted at the outset that the term “bladed disk” is commonlyused to refer to any (blade-containing) rotating or non-rotating part ofan engine or rotating apparatus without necessarily restricting the termto refer to just a disk-shaped rotating part. Thus, a bladed disk canhave externally-attached or integrally-formed blades or any othersuitable rotating protrusions. Also, this rotating mechanism may haveany suitable shape, whether in a disk form or not. Further, the term“bladed disk” may include stators or vanes, which are non-rotatingbladed disks used in gas turbines. Various types of devices such asfans, pumps, turbochargers, compressors, engines, turbines, and thelike, may be commonly referred to as “rotating apparatus.”

[0005]FIG. 1 illustrates a bladed disk 10 which is representative ofthose used in gas turbine engines. One such exemplary gas turbine 12 isillustrated in FIG. 1. Bladed disks used in turbine engines arenominally designed to be cyclically symmetric. If this were the case,then all blades would respond with the same amplitude when excited by atraveling wave. However, in practice, the resonant amplitudes of theblades are very sensitive to small changes in their properties. Thesmall variations that result from the manufacturing process and wearcause some blades to have a significantly higher response and may causethem to fail from high cycle fatigue (HCF). This phenomenon is referredto as the mistuning problem, and has been studied extensively.

[0006]FIG. 2 represents an exemplary selection of nodal diameter modes13-15 in a bladed disk. In the zero nodal diameter mode 13 (part (a) inFIG. 2), all the blades move in phase with one another, while in thehigher nodal diameter modes 14-15, the blades move out of phase. FIG.2(b) illustrates a mode 14 with five (5) nodal diameters, whereas FIG.2(c) illustrates a mode 15 with ten (10) nodal diameters. Thedisplacement of the blades as a function of angular position is given inthese modes by functions sin(nθ) and cos(nθ), where “n” defines thenumber of nodal diameters. For a given value of “n”, the correspondingsine and cosine modes both have the same natural frequency. The onlynodal diameter modes which do not have repeated frequencies are thecases of n=0 and n=N/2, where N is the number of blades on the disk.

[0007]FIG. 3 is an exemplary nodal diameter map 16 of a bladed disk'snatural frequencies. Thus, the natural frequencies of a bladed disk areplotted as a function of the number of nodal diameters in theircorresponding mode. When plotted in this fashion, the frequenciescluster into families of modes. Each family consists of a set of N nodaldiameter modes. Each mode 17-19 at the right in FIG. 3 represents theblade deformation in the corresponding family. Although the relativeamplitudes of the blades varies from one nodal diameter to the next, thedeformation within a given blade remains uniform throughout all modeswithin a given family, at least for families which are isolated infrequency from their neighbors. The blade deformation in the first fewfamilies generally resembles the simple bending and torsion modes of acantilevered plate. Many families of modes have most of their strainenergy stored in the blades. These families appear relatively flat inFIG. 3, because the added strain energy introduced with additional nodaldiameters has a minimal effect on their natural frequency. In contrast,mode families with large strain energy in the disk increase theirfrequency rapidly from one nodal diameter to the next.

[0008] Mistuning can significantly affect the vibratory response of abladed disk. This sensitivity stems from the nature of the eigenvalueproblem that describes a disk's modes and natural frequencies. Aneigenvalue is equal to the square of the natural frequency of a mode.The eigenvalues of a bladed disk are inherently closely spaced due tothe system's rotationally periodic design, as can be seen from the plotin FIG. 3. Therefore, the eigenvectors (mode shapes) of a bladed diskare very sensitive to the small perturbations caused by mistuning. Inthe case of very small mistuning, the blade displacements in the modesare given by distorted sine and cosine waves, while large mistuning canalter the modes to such an extent that the majority of the motion willbe localized to just one or two blades. FIG. 4 illustrates exemplaryforced response tracking plots 20-21 of a tuned bladed disk system (plot20) and the mistuned disk system (plot 21). The plot 21 illustratesblade amplitude magnification caused by mistuning.

[0009] To address the mistuning problem, researchers have developedreduced order models (ROMs) of the bladed disk. These ROMs have thestructural fidelity of a finite element model of the full rotor, whileincurring computational costs that are comparable to that of amass-spring model. In numerical simulations, most published ROMs havecorrelated extremely well with numerical benchmarks. However, somemodels have at times had difficulty correlating with experimental data.These results suggest that the source of the error may lie in theinability to determine the correct input parameters to the ROMs.

[0010] The standard method of measuring mistuning in rotors withattachable blades is to mount each blade in a broach block and measureits natural frequency. The difference of each blade's natural frequencyfrom the mean value is then taken as a measure of the mistuning.However, the mistuning measured through this method may be significantlydifferent from the mistuning present once the blades are mounted on thedisk. This variation in mistuning can arise because each blade'sfrequency is dependent on the contact conditions at the attachment. Notonly may the blade-broach contact differ from the blade-disk contact,but the contact conditions can also vary from slot-to-slot around thewheel or disk. Therefore, to accurately measure mistuning, it isdesirable to develop methods that can infer the mistuning from thevibratory response of the blade-disk assembly as a whole. In addition,many blade-disk structural systems are now manufactured as a singlepiece in which the blades cannot be physically separated from the disk.In the gas turbine industry they are referred to as blisks (for bladeddisks) or IBRs (for integrally bladed rotors). Thus, in the case ofIBRs, the conventional testing methods of separately measuringindividual blade frequencies cannot be applied and, therefore, it isdesirable to develop methods that can infer the properties of theindividual blades from the behavior of the blade-disk assembly as awhole.

[0011] Therefore, to accurately measure mistuning, it is desirable todevelop methods or reduced order models that apply to individual bladesor the blade-disk assembly as a whole. It is further desirable to usethe mistuning values obtained from the newly-devised reduced ordermodels to verify finite element models of the system and also to monitorthe frequencies of individual blades to determine if they have changedbecause of cracking, erosion or other structural changes. It is alsodesirable that the obtained mistuning values can be analyticallyadjusted and used with the reduced order model to predict the vibratoryresponse of the structure (or bladed disk) when it is in use, e.g., whenit is rotating in a gas turbine engine, an industrial turbine, a fan, orany other rotating apparatus.

SUMMARY

[0012] In one embodiment, the present disclosure contemplates a methodthat comprises obtaining a set of vibration measurements that providesfrequency deviation of each blade of a bladed disk system from the tunedfrequency value of the blade and nominal frequencies of the bladed disksystem when tuned; and calculating the mistuned modes and naturalfrequencies of the bladed disk system from the blade frequencydeviations and the nominal frequencies.

[0013] In another embodiment, the present disclosure contemplates amethod that comprises obtaining nominal frequencies of a bladed disksystem when tuned; measuring at least one mistuned mode and naturalfrequency of the bladed disk system; and calculating mistuning of atleast one blade (or blade-disk sector) in the bladed disk system fromonly the nominal frequencies and the at least one mistuned mode andnatural frequency.

[0014] In a further embodiment, the present disclosure contemplates amethod that comprises measuring a set of mistuned modes and naturalfrequencies of a bladed disk system; and calculating mistuning of atleast one blade in the bladed disk system and nominal frequencies of thebladed disk system when tuned by using only the set of mistuned modesand natural frequencies.

[0015] In a still further embodiment, the present disclosurecontemplates a method that comprises obtaining frequency response dataof each blade in a bladed disk system to a traveling wave excitation;transforming data related to spatial distribution of the traveling waveexcitation and the frequency response data; and determining a set ofmistuned modes and natural frequencies of the bladed disk system usingdata obtained from the transformation.

[0016] According to the methodology of the present disclosure, a reducedorder model called the Fundamental Mistuning Model (FMM) is developed toaccurately predict vibratory response of a bladed disk system. FMM maydescribe the normal modes and natural frequencies of a mistuned bladeddisk using only its tuned system frequencies and the frequency mistuningof each blade/disk sector (i.e., the sector frequencies). If the modaldamping and the order of the engine excitation are known, then FMM canbe used to calculate how much the vibratory response of the bladed diskwill increase because of mistuning when it is in use. The tuned systemfrequencies are the frequencies that each blade-disk and blade wouldhave were they manufactured exactly the same as the nominal designspecified in the engineering drawings. The sector frequenciesdistinguish blade-disks with high vibratory response from those with alow response. The FMM identification methods—basic and advanced FMM IDmethods—use the normal (i.e., mistuned) modes and natural frequencies ofthe mistuned bladed disk measured in the laboratory to determine sectorfrequencies as well as tuned system frequencies. Thus, one use of theFMM methodology is to: identify the mistuning when the bladed disk is atrest, to extrapolate the mistuning to engine operating conditions, andto predict how much the bladed disk will vibrate under the operating(rotating) conditions.

[0017] In one embodiment, the normal modes and natural frequencies ofthe mistuned bladed disk are directly determined from the disk'svibratory response to a traveling wave excitation in the engine. Thesemodes and natural frequency may then be input to the FMM ID methodologyto monitor the sector frequencies when the bladed disk is actuallyrotating in the engine. The frequency of a disk sector may change if theblade's geometry changes because of cracking, erosion, or impact with aforeign object (e.g., a bird). Thus, field calibration and testing ofthe blades (e.g., to assess damage from vibrations in the engine) may beperformed using traveling wave analysis and FMM ID methods together.

[0018] The FMM software (containing FMM ID methods) may receive therequisite input data and, in turn, predict bladed disk system'smistuning and vibratory response. Because the FMM model can be generatedcompletely from experimental data (e.g., using the advanced FMM IDmethod), the tuned system frequencies from advanced FMM ID may be usedto validate the tuned system finite element model used by industry.Further, FMM and FMM ID methods are simple, i.e., no finite element massor stiffness matrices are required. Consequently, no special interfacesare required for FMM to be compatible with a finite element model.

BRIEF DESCRIPTION OF THE DRAWINGS

[0019] For the present disclosure to be easily understood and readilypracticed, the present disclosure will now be described for purposes ofillustration and not limitation, in connection with the followingfigures, wherein:

[0020]FIG. 1 illustrates a bladed disk which is representative of thoseused in gas turbine engines;

[0021]FIG. 2 represents an exemplary selection of nodal diameter modesin a bladed disk;

[0022]FIG. 3 is an exemplary nodal diameter map of a bladed disk'snatural frequencies;

[0023]FIG. 4 illustrates exemplary forced response tracking plots of atuned bladed disk system and the mistuned disk system;

[0024]FIG. 5 illustrates near equivalence of sector modes from variousnodal diameters;

[0025]FIG. 6 illustrates an exemplary three dimensional (3D) finiteelement model of a high pressure turbine (HPT) blade-disk sector;

[0026]FIG. 7 shows tuned system frequencies of the first families ofmodes of the blade-disk sector modeled in FIG. 6;

[0027]FIG. 8 illustrates the tuned frequencies of the fundamental familyof modes of the system modeled in FIG. 6, along with the frequencies asdetermined by ANSYS® software and the best-fit mass-spring model;

[0028] FIGS. 9(a) and (b) depict representative results of using FMMwith a realistic mistuned bladed disk modeled in FIG. 6;

[0029] FIGS. 10(a) and (b) show a representative case of the bladeamplitudes as a function of excitation frequency for a 7^(th) engineorder excitation predicted by the mass-spring model, ANSYS® software,and FMM;

[0030]FIG. 11 illustrates the leading edge blade tip displacements forthe third family of modes shown in FIG. 7;

[0031] FIGS. 12(a)-(c) illustrate FMM and ANSYS® software predictions ofblade amplitude as a function of excitation frequency for a 2^(nd)engine order excitation of 2^(nd), 3^(rd), and 4^(th) familiesrespectively;

[0032] FIGS. 13(a)-(c) illustrate FMM and ANSYS® software predictions ofblade amplitude as a function of excitation frequency for a 7^(th)engine order excitation of 2^(nd), 3^(rd), and 4^(th) familiesrespectively;

[0033]FIG. 14 represents an exemplary finite element model of a twentyblade compressor;

[0034]FIG. 15 illustrates the natural frequencies of the tunedcompressor modeled in FIG. 14;

[0035]FIG. 16 shows the comparison between the sector mistuningcalculated directly by finite element simulations of each mistunedblade/sector and the mistuning identified by basic FMM ID;

[0036]FIG. 17 schematically illustrates a rotor 29 with exaggeratedstagger angle variations as viewed from above;

[0037]FIG. 18 shows a representative mistuned mode caused by staggerangle mistuning of the rotor in FIG. 14;

[0038]FIG. 19 illustrates a comparison of mistuning determination frombasic FMM ID and the variations in the stagger angles;

[0039]FIG. 20 depicts a comparison of mistuning predicted using advancedFMM ID with that obtained using the finite element analysis (FEA);

[0040]FIG. 21 shows a comparison of the tuned frequencies identified byadvanced FMM ID and those computed directly with the finite elementmodel;

[0041]FIG. 22 illustrates an exemplary setup to measure transferfunctions of test rotors and also to verify various FMM ID methods;

[0042]FIG. 23 illustrates natural frequencies of a test compressor withno mistuning;

[0043]FIG. 24 illustrates a typical transfer function from a testcompressor obtained using the test setup shown in FIG. 22;

[0044]FIG. 25 illustrates a comparison of mistuning from each FMM IDmethod with benchmark results for a test rotor SN-1;

[0045]FIG. 26 shows a comparison of tuned system frequencies for thetest rotor SN-1 from advanced FMM ID and the finite element model (FMM)using ANSYS® software;

[0046]FIGS. 27 and 28 are similar to FIGS. 25 and 26, respectively, butillustrate the identified mistuning and tuned system frequencies for adifferent test rotor SN-3;

[0047] FIGS. 29(a) and (b) show a comparison, for rotors SN-1 and SN-3respectively, of the mistuning identified by FMM ID with the values frombenchmark results from geometric measurements;

[0048]FIG. 30 illustrates a comparison of tuned system frequencies fromadvanced FMM ID and ANSYS® software for torsion modes of rotors SN-1 andSN-3;

[0049]FIG. 31(a) depicts FMM-based forced response data, whereas FIG.31(b) depicts the experimental forced response data;

[0050] FIGS. 32(a) and (b) respectively show relative blade amplitudesat forced response resonance for the resonant peaks labeled {circle over(1)} and {circle over (2)} in FIG. 31(a);

[0051]FIG. 33 depicts cumulative probability function plots of peakblade amplitude for a nominally tuned and nominally mistuned compressor;

[0052]FIG. 34 shows mean and standard deviations of each sector'smistuning for a nominally mistuned compressor;

[0053]FIG. 35 illustrates a lumped parameter model of a rotating blade;

[0054]FIG. 36 shows a comparison of mistuning values analyticallyextrapolated to speed with an FEA (finite element analysis) benchmark;

[0055]FIG. 37 illustrates the effect of centrifugal stiffening on tunedsystem frequencies;

[0056]FIG. 38 illustrates the effect of centrifugal stiffening onmistuning;

[0057]FIG. 39 depicts frequency response of blades to a six engine orderexcitation at 40,000 RPM rotational speed;

[0058] FIGS. 40(a), (b) and (c) show a comparison of the representativemode shape extracted from the traveling wave response data withbenchmark results for a stationary benchmark;

[0059]FIG. 41 depicts comparison of the natural frequencies extractedfrom the traveling wave response data with the benchmark results for thestationary benchmark of FIG. 40;

[0060]FIG. 42 shows a calibration curve relating the effect of a unitmass on a sector's frequency deviation in a stationary benchmark;

[0061]FIG. 43 shows the comparison between the mass mistuning identifiedthrough traveling wave FMM ID with the values of the actual massesplaced on each blade tip;

[0062] FIGS. 44(a) and (b) show tracking plots of blade amplitudes as afunction of excitation frequency for two different acceleration rates;

[0063] FIGS. 45(a) and (b) illustrate the comparison of the mistuningdetermined through the traveling wave system identification method withbenchmark values for two different acceleration rates; and

[0064]FIG. 46 illustrates an exemplary process flow depicting variousblade sector mistuning tools discussed herein.

DETAILED DESCRIPTION

[0065] Reference will now be made in detail to some embodiments of thepresent disclosure, examples of which are illustrated in theaccompanying drawings. It is to be understood that the figures anddescriptions of the present disclosure included herein illustrate anddescribe elements that are of particular relevance to the presentdisclosure, while eliminating, for the sake of clarity, other elementsfound in typical bladed disk systems, engines, or rotating devices. Itis noted here that the although the discussion given below is primarilywith reference to a blade-disk sector, the principles given below can beequally applied to just the blade portion of the blade-disk sector ascan be appreciated by one skilled in the art. Therefore, the terms“blade” and “blade-disk sector” have been used interchangeably in thediscussion below, and no additional discussion of the blade-onlyapplication is presented herein.

[0066] [1] Deriving the Fundamental Mistuning Model (FMM)

[0067] The more general form of the modal equation for the FundamentalMistuning Model (FMM), derived below, is applicable to rotating, bladedapparatus. The generalized FMM formulation differs in two ways from theoriginal FMM derivation described in the paper by D. M. Feiner and J. H.Griffin titled “A Fundamental Model of Mistuning for a Single Family ofModes,” appearing in the Proceedings of IGTI, ASME Turbo Expo 2002 (Jun.3-6, 2002), Amsterdam, The Netherlands. This paper is incorporatedherein by reference in its entirety. First, the following derivation nolonger approximates the tuned system frequencies by their average value.This allows for a much larger variation among the tuned frequencies.Second, rather than using the blade-alone mode as an approximation ofthe various nodal diameter sector modes, a representative mode of asingle blade-disk sector is used below. Consequently, the approach nowincludes the disk portion of the mode shape, and thus allows for morestrain energy in the disk. Although mistuning may be measured as apercent deviation in the blade-alone frequency (as in the abovementioned paper), in the following discussion mistuning is measured as apercent deviation in the frequency of each blade-disk sector. The sectorfrequency deviations not only capture mistuning in the blade, but canalso capture mistuning in the disk as well as variations in the ways theblades are attached to the disk.

[0068] In the discussion below, section 1.1 describes how the subset ofnominal modes (SNM) approach is used to reduce the order of the mistunedfree-response equations and formulates the problem in terms of reducedorder sector matrices, section 1.2 relates the sector matrices tomistuned sector frequencies, and section 1.3 simplifies the resultingmathematical expressions.

[0069] 1.1 Reduction of Order

[0070] Consider a mistuned, bladed disk in the absence of an excitation.The order of its equation of motion may be reduced through a subset ofnominal modes (SNM) approach. The resulting reduced order equation canbe written as (see, for example, the discussion in the above mentionedFeiner-Griffin paper):

[(Ω°² +Δ{circumflex over (K)})−ω_(j) ²(I+Δ{circumflex over (M)})]{rightarrow over (β)}_(j)=0  (1)

[0071] Ω°² is a diagonal matrix of the tuned system eigenvalues (aneigenvalue is equal to the square of the natural frequency of a mode),and I is the identity matrix. Δ{circumflex over (K)} and Δ{circumflexover (M)} are the variations in the modal stiffness and modal massmatrices caused by stiffness and mass mistuning. The vector {right arrowover (β)}_(j) contains weighting factors that describe the j^(th)mistuned mode as a limited sum of tuned modes, i.e.,

{right arrow over (φ)}_(j)=Φ°{right arrow over (β)}_(j)  (2)

[0072] where Φ° is a matrix whose columns are the tuned system modes.

[0073] Note that to first order, (I+Δ{circumflex over(M)})⁻¹≈(I−Δ{circumflex over (M)}). Thus by pre-multiplying (1) by(I+Δ{circumflex over (M)})⁻ and keeping only first order terms, theexpression becomes

(Λ°+Â){right arrow over (β)}_(j)=ω_(j) ²{right arrow over (β)}_(j)  (3)

[0074] where

Â=Δ{circumflex over (K)}−Δ{circumflex over (M)}Ω° ²  (4)

[0075] The next section relates the matrix Â to the frequency deviationsof the mistuned sectors.

[0076] 1.2 Relating Mistuning to Sector Frequency Deviations

[0077] Relating Â to frequency deviations of the sectors is a three-stepprocess. First, the mistuning matrix is express in terms of the systemmode shapes of an individual sector. Then, the system sector modes arerelated to the corresponding mode of a single, isolated sector. Finally,the resulting sector-mode terms in Â are expressed in terms of thefrequency deviations of the sectors.

[0078] 1.2.1 Relating Mistuning to System Sector Modes

[0079] Consider the mistuning matrix, Â, in equation (4). This matrixcan be expressed as a sum of the contributions from each mistunedsector. $\begin{matrix}{\hat{A} = {\sum\limits_{s = 0}^{N - 1}\quad {\hat{A}}^{(s)}}} & (5)\end{matrix}$

[0080] where the superscript denotes that the mistuning corresponds tothe s^(th) sector. The expression for a single element of Â^((s)) is$\begin{matrix}{{\hat{A}}_{mn}^{(s)} = {{{\overset{}{\varphi}}_{m}^{{\circ {(s)}}H}( {{\Delta \quad K^{(s)}} - {\omega_{n}^{\circ 2}\Delta \quad M^{(s)}}} )}{\overset{}{\varphi}}_{n}^{\circ {(s)}}}} & (6)\end{matrix}$

[0081] where ΔK^((s)) and ΔM^((s)) are the physical stiffness and massperturbations of the s^(th) sector. The modes {right arrow over(φ)}_(m)°^((s)) and {right arrow over (φ)}_(n)°^((s)) are the portionsof the m^(th) and n^(th) columns of Φ° which describe the s^(th)sector's motion. The term ω_(n)°² is the nth diagonal element of Ω°².Equation (6) relates the mistuning to the system sector modes. In thenext section, these modes are related to the mode of a single isolatedblade-disk sector.

[0082] 1.2.2 Relating System Sector Modes to an Average Sector Mode

[0083] The tuned modes in equation (6) are expressed in a complextraveling wave form. Thus, the motion of the s^(th) sector can berelated to the motion of the 0^(th) sector by a phase shift. Thus,equation (6) can be restated as $\begin{matrix}{{\hat{A}}_{mn}^{(s)} = {^{{{is}{({n - m})}}\frac{2\pi}{N}}{{\overset{}{\varphi}}_{m}^{{\circ {(s)}}H}( {{\Delta \quad K^{(s)}} - {\omega_{n}^{\circ 2}\Delta \quad M^{(s)}}} )}{\overset{}{\varphi}}_{n}^{\circ {(0)}}}} & (7)\end{matrix}$

[0084] Because the tuned modes used in the SNM formulation are anisolated family of modes, the sector modes of all nodal diameters looknearly identical as can be seen from FIG. 5, which illustrates nearequivalence of sector modes from various nodal diameters. Therefore, onecan approximate the various sector modes by an average sector mode.Applying the average sector mode approximation for the system sectormodes in equation (7), Â_(mn) ^((s)) can be written as $\begin{matrix}{{\hat{A}}_{m\quad n}^{(s)} = {( \frac{\omega_{m}^{\circ}\omega_{n}^{\circ}}{\omega_{\psi}^{\circ 2}} ){^{\quad {s{({n - m})}}\frac{2\quad \pi}{N}}\lbrack {{{\overset{arrow}{\psi}}^{\circ {(0)}^{H}}( {{\Delta \quad K^{(s)}} - {\omega_{n}^{\circ 2}\Delta \quad M^{(s)}}} )}{\overset{arrow}{\psi}}^{\circ {(0)}}} \rbrack}}} & (8)\end{matrix}$

[0085] where {right arrow over (ψ)}°⁽⁰⁾ is the average tuned systemsector mode, and ω_(ψ)° is its natural frequency. In practice, {rightarrow over (ψ)}°⁽⁰⁾ can be taken to be the median modal diameter mode.The factor (ω_(m)°ω_(n)°)/(ω_(ψ)°²) scales the average sector mode termsso that they have approximately the same strain energy as the sectormodes they replace.

[0086] 1.2.3 Introduction of Sector Frequency Deviation

[0087] The deviation in a sector frequency quantity may be used tomeasure mistuning. To understand this concept, consider an imaginary“test” rotor. In the test rotor every sector is mistuned in the samefashion, so as to match the mistuning in the sector of interest. Sincethe test rotor's mistuning is cyclically symmetric, its mode shapes arevirtually identical to those of the tuned system. However, there will bea shift in the tuned system frequencies. For small levels of mistuning,the frequency shift is nearly the same in all of the tuned system modesand can be approximated by the fractional change in the frequency of themedian nodal diameter mode. This may typically be the case for anisolated family of modes in which the strain energy is primarily in theblades. If there is a significant amount of strain energy in the diskthen the frequency of the modes may change significantly as a functionof nodal diameter and the modes may not be isolated (i.e., the modes maycover such a broad frequency range that they may interact with otherfamilies of modes). However, in the following, the fractional shift inthe median nodal diameter's frequency is taken as a measure of mistuningand is defined as the sector frequency deviation.

[0088] The bracketed terms of in equation (8) are related to thesefrequency deviations in the following manner. Consider a bladed diskthat is mistuned in a cyclic symmetric fashion, i.e., each sectorundergoes the same mistuning. Its free-response equation of motion isgiven by the expression

[(K°+ΔK)−ω_(n) ²(M°+ΔM)]{right arrow over (φ)}_(n)=0  (9)

[0089] Take the mode {right arrow over (φ)}_(n) to be the mistunedversion of the tuned median nodal diameter mode, {right arrow over(ψ)}°. Here, {right arrow over (ψ)}° is the full system mode counterpartof the average sector mode {right arrow over (ψ)}°⁽⁰⁾. Because mistuningis symmetric, the tuned and mistuned versions of the mode are nearlyidentical. Substituting {right arrow over (ψ)}° for {right arrow over(φ)}_(n) and pre-multiplying by {right arrow over (ψ)}°^(H) yields,

(ω_(ψ)°²+{right arrow over (ψ)}^(°H) ΔK{right arrow over (ψ)}^(°))−ω_(n)²(1+{right arrow over (ψ)}^(°H) ΔM{right arrow over (ψ)}^(°))=0  (10)

[0090] These terms may be rearranged to isolate the frequency terms,

{right arrow over (ψ)}^(°H)(ΔK−ω _(n) ² ΔM){right arrow over(ψ)}^(°)=ω_(j) ²−ω_(ψ)°²  (11)

[0091] Because the mistuning is symmetric, each sector contributesequally to equation (11). Thus, the contribution from the 0^(th) sectoris, $\begin{matrix}{{{{\overset{arrow}{\psi}}^{{\circ {(0)}}H}( {{\Delta \quad K} - {\omega_{n}^{2}\Delta \quad M}} )}{\overset{arrow}{\psi}}^{\circ {(0)}}} = {\frac{1}{N}( {\omega_{j}^{2} - \omega_{\psi}^{\circ 2}} )}} & (12)\end{matrix}$

[0092] By factoring the frequency terms on the right-hand side ofequation (12), it can be shown that $\begin{matrix}{{{{\overset{arrow}{\psi}}^{{\circ {(0)}}H}( {{\Delta \quad K} - {\omega_{n}^{2}\Delta \quad M}} )}{\overset{arrow}{\psi}}^{\circ {(0)}}} \approx \frac{2\quad \omega_{\psi}^{\circ 2}{\Delta\omega}_{\psi}}{N}} & (13)\end{matrix}$

[0093] where Δω_(ψ) is the fractional change in {right arrow over (ψ)}'snatural frequency due to mistuning, given byΔω_(ψ)=(ω_(ψ)−ω_(ψ)°)/ω_(ψ)°. Note that, by definition, Δω_(ψ) is asector frequency deviation. Equation (13) can be substituted for thebracketed terms of equation (8), resulting in an expression that relatesthe elements of the sector “s” mistuning matrix to that sector'sfrequency deviation, $\begin{matrix}{{\hat{A}}_{m\quad n}^{(s)} = {\frac{2\omega_{m}^{\circ}\omega_{n}^{\circ}}{N}^{\quad {s{({n - m})}}\frac{2\quad \pi}{N}}{\Delta\omega}_{\psi}^{(s)}}} & (14)\end{matrix}$

[0094] where the superscript on Δω_(ψ) is introduced to indicate thatthe frequency deviation corresponds to the s^(th) sector. These sectorcontributions may be summed to obtain the elements of the mistuningmatrix, $\begin{matrix}{{\hat{A}}_{m\quad n} = {2\omega_{m}^{\circ}{\omega_{n}^{\circ}\lbrack {\frac{1}{N}{\sum\limits_{s = 0}^{N - 1}\quad {^{\quad s\quad p\frac{2\quad \pi}{N}}\Delta \quad \omega_{\psi}^{(s)}}}} \rbrack}}} & (15)\end{matrix}$

[0095] 1.3 The Simplified Form of the Fundamental Mistuning Model ModalEquation

[0096] The bracketed term in equation (15) is the discrete Fouriertransform (DFT) of the sector frequency deviations. If one uses thedummy variable p to replace the quantity (n−m) in equation (15), thenthe p^(th) DFT of the sector frequency deviations is given by$\begin{matrix}{{\overset{\_}{\omega}}_{p} = \lbrack {\frac{1}{N}{\sum\limits_{s = 0}^{N - 1}\quad {^{\quad s\quad p\frac{2\quad \pi}{N}}{\Delta\omega}_{\psi}^{(s)}}}} \rbrack} & (16)\end{matrix}$

[0097] where {overscore (ω)}_(p) denotes the p^(th) DFT . Bysubstituting equation (16) into equation (15), Â may be expressed in thesimplified matrix form

Â=2Ω°{overscore (Ω)}Ω°  (17)

[0098] where $\begin{matrix}{\overset{\_}{\Omega} = \begin{bmatrix}{\overset{\_}{\omega}}_{0} & {\overset{\_}{\omega}}_{1} & \cdots & {\overset{\_}{\omega}}_{N - 1} \\{\overset{\_}{\omega}}_{N - 1} & {\overset{\_}{\omega}}_{0} & \cdots & {\overset{\_}{\omega}}_{N - 2} \\\vdots & \vdots & \quad & \vdots \\{\overset{\_}{\omega}}_{1} & {\overset{\_}{\omega}}_{2} & \cdots & {\overset{\_}{\omega}}_{0}\end{bmatrix}} & (18)\end{matrix}$

[0099] {overscore (Ω)} is a matrix which contains the discrete Fouriertransforms of the sector frequency deviations. Note that {overscore (Ω)}has a circulant form, and thus contains only N distinct elements.{overscore (Ω)}° is a diagonal matrix of the tuned system frequencies.

[0100] Substituting equation (17) into equation (3) produces the mostbasic form of the eigenvalue problem that may be solved to determine themodes and natural frequencies of the mistuned system.

(Ω°²+2Ω°{overscore (Ω)}Ω°){right arrow over (β)}_(j)=ω_(j) ²{right arrowover (β)}_(j)  (19)

[0101] Equations (18) and (19) represent the functional form of theFundamental Mistuning Model. Here, Ω°² is a diagonal matrix of thenominal system eigenvalues, ordered in accordance with the followingequation. $\begin{matrix}\begin{Bmatrix}{\overset{arrow}{\varphi}}_{0}^{\circ {(s)}} & {\overset{arrow}{\varphi}}_{1}^{\circ {(s)}} & {\overset{arrow}{\varphi}}_{2}^{\circ {(s)}} & \cdots & {\overset{arrow}{\varphi}}_{\frac{N}{2}}^{\circ {(s)}} & {\overset{arrow}{\varphi}}_{\frac{N}{2} + 1}^{\circ {(s)}} & \cdots & {\overset{arrow}{\varphi}}_{N - 1}^{\circ {(s)}} \\(0) & ( {1B} ) & ( {2B} ) & \cdots & ( \frac{N}{2} ) & ( {( {\frac{N}{2} - 1} )F} ) & \cdots & ( {1F} )\end{Bmatrix} & (20)\end{matrix}$

[0102] where the second row in equation (20) indicates the nodaldiameter and direction of the corresponding mode that lies above it. “B”denotes a backward traveling wave, defined as a mode with a positivephase shift from one sector to the next, and “F” denotes a forwardtraveling wave, defined as a mode with a negative phase shift from onesector to the next. Note that the modes are numbered from 0 to N−1.

[0103] As mentioned before, the eigenvalues are equal to the squares ofthe natural frequencies of the tuned system. This Ω°² matrix containsall the nominal system information required to calculate the mistunedmodes. Note that the geometry of the system does not directly enter intothis expression. The term representing mistuning in equation (1),2Ω°{overscore (Ω)}Ω°, is a simple circulant matrix that contains thediscrete Fourier transforms of the blade frequency deviations, pre- andpost-multiplied by the tuned system frequencies.

[0104] The eigenvalues of equation (19) are the squares of the mistunedfrequencies, and the eigenvectors define the mistuned mode shapesthrough equation (2). Because the tuned modes have been limited to asingle family and appear in Φ° in a certain order, one can approximatelycalculate the distortion in the mistuned mode shapes without knowinganything specific about Φ°. The reason for this is the assumption thatall of the tuned system modes on the zero^(th) sector look nearly thesame, i.e. {right arrow over (φ)}_(m) ^(°(0))≈{right arrow over (φ)}_(n)^(°(0)). Further, when the tuned system modes are written in complex,traveling wave form, the amplitudes of every blade in a mode are thesame, but each blade has a different phase. Therefore, the part of themode corresponding to the s^(th) sector can be written in terms of thesame mode on the 0^(th) sector, multiplied by an appropriate phaseshift, i.e., $\begin{matrix}{{\overset{arrow}{\varphi}}_{n}^{\circ {(s)}} = {^{\quad s\quad n\frac{2\quad \pi}{N}}{\overset{arrow}{\varphi}}_{n}^{\circ {(0)}}}} & (21)\end{matrix}$

[0105] where i={square root}{square root over (−1)}. Equation (21)implies that if the j^(th) mistune mode is given by {right arrow over(β)}_(j)=[β_(j0), β_(j1) . . . β_(j,N−1)]^(T) then the physicaldisplacements of the n^(th) blade in this mode proportional to$\begin{matrix}{x_{n} = {\sum\limits_{m = 0}^{N - 1}\quad {\beta_{j\quad m}^{\quad m\quad n\frac{2\quad \pi}{N}}}}} & (22)\end{matrix}$

[0106] 1.4 Numerical Results

[0107] A computer program was written to implement the theory presentedin sections (1.1) through (1.3). The program also incorporated a simplemodal summation algorithm to calculate the bladed disk's forcedresponse. The modal summation assumed constant modal damping. The basicmodal summation algorithm was chosen to benchmark the forced responsebecause a similar algorithm may be used as an option in the commercialfinite element analysis ANSYS® software, which was used as a benchmark.It is observed, however, that FMM may be used with more sophisticatedmethods for calculating the forced response, such as the state-spaceapproach used in a subset of nominal modes (SNM) analysis discussed inYang M.-T. and Griffin, J. H., 2001, “A Reduced Order Model of MistuningUsing a Subset of Nominal Modes,” Journal of Engineering for GasTurbines and Power, 123(4), pp. 893-900.

[0108] It is noted that when a beam-like blade model is used (tominimize the computational cost), FMM could accurately calculate abladed disk's mistuned response based on only sector frequencydeviations, without regard for the physical cause of the mistuning.However, in the discussion below, a more realistic geometry is analyzedusing FMM.

[0109]FIG. 6 illustrates an exemplary three dimensional (3D) finiteelement model 22 of a high pressure turbine (HPT) blade-disk sector.There were 24 sectors in the full system. This model was developed byapproximating the features of an actual turbine blade and provided areasonable test of FMM's ability to represent a realistic bladegeometry. FIG. 7 shows tuned system frequencies of the first families ofmodes of the blade-disk system modeled in FIG. 6. As can be seen fromFIG. 7, the system of FIG. 6 had an isolated first bending family ofmodes with closely spaced frequencies. As a benchmark, a finite elementanalysis was performed of the full mistuned rotor using the ANSYS®software. The bladed disk was mistuned by randomly varying the elasticmoduli of the blades with a standard deviation that was equal to 1.5% ofthe tuned system's elastic modulus.

[0110] Then, an equivalent mass-spring model was constructed with onedegree-of-freedom per sector as described in Rivas-Guerra, A. J., andMignolet, M. P., 2001, “Local/Global Effects of Mistuning on the ForcedResponse of Bladed Disks,” ASME Paper 2001-GT-0289, International GasTurbine Institute Turbo Expo, New Orleans, La. Each mass was set tounity and the stiffness parameters were obtained through a least squaresfit of the tuned natural frequencies. FIG. 8 illustrates the tunedfrequencies of the fumdamental family of modes of the system modeled inFIG. 6, along with the frequencies of the ANSYS® software and thebest-fit mass-spring model. It is noted that while the mass-spring modelwas able to capture the higher nodal diameter frequencies fairly well,it had great difficulty with the low nodal diameter frequencies. Thisdiscrepancy arises because the natural frequencies of the singledegree-of-freedom mass-spring system have the form

ω_(n) ={square root}{square root over ([k+4k_(c) sin²(nπ/N)]/m)}  (23)

[0111] where m is the blade mass, k and k_(c) are the base stiffness andcoupling stiffness, n is the nodal diameter of the mode, and N is thenumber of blades. However, the actual frequencies of the finite elementmodel have a significantly different shape when plotted as a function ofnodal diameters. In contrast, FMM takes the actual finite elementfrequencies as input parameters, and therefore it matches the tunedsystem's frequencies exactly.

[0112] The mass-spring model was then mistuned by adjusting the basestiffness of the blades to correspond to the modulus changes used in thefinite element model. The mistuned modes and forced response were thencalculated with both FMM and the mass-spring model, and compared withthe finite element results using the ANSYS® software. FIGS. 9(a) and (b)depict representative results of using FMM with a realistic mistunedbladed disk modeled in FIG. 6. As can be seen from FIG. 9(a), themistuned frequencies predicted by FMM and ANSYS® software were quitesimilar. However the mass-spring model had some significantdiscrepancies, especially in the low frequency modes. FMM and ANSYS®software also predicted essentially the same mistuned mode shapes as canbe seen from FIG. 9(b). In contrast, the mass-spring model performedpoorly when matching the finite element mode shapes, even on modes whosefrequencies were accurately predicted. For example, the mode plotted inFIG. 9(b) corresponded to the 18^(th) frequency. From FIG. 9(a), it isseen that the mass-spring model accurately predicted this frequency.However, it is clear from FIG. 9(b) that the mass-spring model still dida poor job of matching the finite element mode shapes.

[0113] The predicted modes were then summed to obtain the system'sforced response to various engine order excitations. As noted before,gas turbine engines are composed of a series of bladed disks (see, forexample, FIG. 1). When a bladed disk is operating in an engine, it issubjected to pressure loading from the flow field which excites theblades. As the flow progresses through the engine, it passes oversupport struts, inlet guide vanes, and other stationary structures whichcause the pressure to vary circumferentially. Therefore, the excitationforces are periodic in space when considered from a stationary referenceframe. As a periodic excitation, the pressure variations can bespatially decomposed into a Fourier series. Each harmonic componentdrives the system with a traveling wave at a frequency given by theproduct of its harmonic number and the rotation speed. The harmonicnumber of the excitation is typically referred to as the Engine Order,and corresponds physically to the number of excitation periods perrevolution. Each of the engine order excitations may be generallyconsidered separately, because they drive the system at differentfrequencies.

[0114] FIGS. 10(a) and (b) show a representative case of the bladeamplitudes as a function of excitation frequency for a 7^(th) engineorder excitation predicted by the mass-spring model, ANSYS® software,and FMM. For clarity, only the high responding, median responding, andlow responding blades are plotted. It is again seen that the mass-springmodel provided a poor prediction of the system's forced response.However, the results from FMM agreed well with those computed by ANSYS®software, as shown in FIG. 10(b). The prediction by FMM of the highestblade response differed from that predicted by ANSYS® software by only1.6%. Thus, FMM may be used to provide accurate predictions of the modeshapes and forced response of a turbine blade with a realistic geometry.

[0115] 1.5 Other Considerations

[0116] From the foregoing discussion, it is seen that the FundamentalMistuning Model was derived from the Subset of Nominal Modes theory byapplying three simplifying assumptions: only a single, isolated familyof modes is excited; the strain energy of that family's modes isprimarily in the blades; and the family's frequencies are closelyspaced. In addition, one corollary of these assumptions is that theblade's motion looks very similar among all modes in the family. Asdemonstrated in the previous section, FMM works quite well when theseassumptions are met. However, these ideal conditions are usually foundonly in the fundamental modes of a rotor. The higher frequency familiesare often clustered close together, have a significant amount of strainenergy in the disk, and span a large frequency range. Furthermore,veerings are quite common, causing a family's modes to changesignificantly from one nodal diameter to the next. Therefore, there maybe situations where FMM may not work effectively in high frequencyregions.

[0117] The realistic HPT model of FIG. 6 may be used to further studyFMM's performance, without the need carefully assign modes to families.Therefore, some crossings shown in FIG. 7 may in fact be veerings.However, because such errors are easily made in practice, it is usefulto include them in the study. For reference, four mode families arenumbered along the right side of FIG. 7. It is noted that except for thefundamental modes, the families (in FIG. 7) undergo a significantfrequency increase between 0 and 6 nodal diameters. The steep slopes inthis region suggest that the modes have a large amount of strain energyin the disk. Furthermore, the high modal density in this area makes itlikely that some modes were assigned to the wrong family. Therefore, themodes of a single family may likely change significantly from one nodaldiameter to the next. To show this behavior, FIG. 11 illustrates theleading edge blade tip displacements for the third family of modes shownin FIG. 7. FIG. 11 shows how the circumferential (θ) and out-of-plane(z) motion of the blade tip's leading edge changes from the 0 nodaldiameter mode to the 12 nodal diameter mode. Observe that the θ and zcomponents of the mode shape change significantly between 0 and 6 nodaldiameters. In such case, the assumptions of FMM are violated throughoutthese low nodal diameter regions. Thus, FMM may not accurately predictthe mistuned frequencies or shapes of these modes. As a result, FMM maynot provide accurate forced response predictions when these modes areheavily excited.

[0118] To illustrate FMM performance in such situations, FMM was used topredict the forced response of families 2, 3, and 4 to a 2^(nd)engine-order excitation because that engine order would primarily excitethe low nodal diameter modes of each family, and those modes violate theassumptions of FMM. The FMM predictions were compared with finiteelement results calculated in ANSYS® software. FIGS. 12(a)-(c)illustrate FMM and ANSYS® software predictions of blade amplitude as afunction of excitation frequency for a 2^(nd) engine order excitation of2^(nd), 3^(rd), and 4^(th) families respectively. For clarity, each plotin FIGS. 12(a)-(c) shows only the low responding blade, the medianresponding blade, and the high responding blade. As expected, the FMMresults differed significantly from the ANSYS® software response in bothpeak amplitudes and overall shape of the response. Thus, when a modelies in a region where there is uncertainty as to what family a modebelongs, veering, or high slopes on the frequency vs. nodal diameterplot, FMM may not always accurately predict its mistuned frequency ormode shape. That is, FMM may not work effectively for engine orders thatexcite modes in these regions.

[0119] However, there are regions in high frequency modes where FMM mayperform quite well. It is seen from the Frequency vs. Nodal Diameterplot in FIG. 7 that the slopes over the high nodal diameter regions arevery small, indicating that the modes have most of their strain energyin the blades. Furthermore, the flat regions are well isolated fromother families of modes. Therefore, the blade's motion is very similarfrom one nodal diameter to the next. This can be seen in FIG. 11, whichindicates that the θ and z components of the blade tip motion remainfairly constant over the higher nodal diameter regions. In that case,the FMM assumptions are satisfied for these high nodal diameter modes,and FMM may capture the physical behavior of these modes better than itdid for low engine orders.

[0120] To illustrate FMM performance in the situation described in thepreceding paragraph, FMM was used to predict the forced response offamilies 2, 3, and 4 to a 7^(th) engine order excitation. The FMMresults were compared against a finite element benchmark performed inANSYS®O software. FIGS. 13(a)-(c) illustrate FMM and ANSYS® softwarepredictions of blade amplitude as a function of excitation frequency fora 7^(th) engine order excitation of 2^(nd), 3^(rd), and 4^(th) familiesrespectively. For clarity, each plot in FIGS. 13(a)-(c) shows only thelow responding blade, the median responding blade, and the highresponding blade. In all three cases in FIG. 13, the FMM predictionscaptured the overall shape of the response curves as well as the peakamplitudes to within 6% of the ANSYS® software performance. Therefore,for this test case, it is observed that when a mode lies in a flatregion at the upper end of a Frequency vs. Nodal Diameter plot, itsresponse can be reasonably well predicted by FMM.

[0121] [2] System Identification Methods

[0122] It is seen from the discussion hereinbefore that the FundamentalMistuning Model provides a simple, but accurate method for assessing theeffect of mistuning on forced response, generally in case of an isolatedfamily of modes. However, FMM can be used to derive more complex reducedorder models to analyze mistuned response in regions of frequencyveering, high modal density and cases of disk dominated modes. Thesecomplex models may not necessarily be limited to an isolated family ofmodes.

[0123] The following description of system identification is based theFundamental Mistuning Model. As a result, the FMM based identificationmethods (FMM ID) (discussed below) may be easy to use and may requirevery little analytical information about the system, e.g., no finiteelement mass or stiffness matrices may be necessary. There are two formsof FMM ID methods discussed below: a basic version of FMM ID thatrequires some information about the system properties, and a somewhatmore advanced version that is completely experimentally based. The basicFMM ID requires the nominal frequencies of the tuned system as input.The nominal frequencies of the tuned system (i.e., natural frequenciesof a tuned system with each sector being identical) may be calculatedusing a finite element analysis of a single blade-disk sector withcyclic symmetric boundary conditions applied to the disk. Then, given(experimental) measurements of a limited number of mistuned modes andfrequencies, basic FMM ID equations solve for the mistuned frequency ofeach sector. It is noted that the modes required in basic FMM ID are thecircumferential modes that correspond to the tip displacement of eachblade around the wheel or disk.

[0124] The advanced form of FMM ID uses (experimental) measurements ofsome mistuned modes and frequencies to determine all of the parametersin FMM, i.e. the frequencies that the system would have if it were tunedas well as the mistuned frequency of each sector. Thus, the tuned systemfrequencies determined from the second method (i.e., advanced FMM ID)can also be used to validate finite element models of the nominalsystem.

[0125] 2.1 Basic FMM ID Method

[0126] As noted before, the basic method uses tuned system frequenciesalong with measurements of the mistuned rotor's system modes andfrequencies to infer mistuning.

[0127] 2.1.1 Inversion of FMM Equation

[0128] The FMM eigenvalue problem is given by equation (19), which isreproduced below.

(Ω°²+2Ω°{overscore (Ω)}Ω°){right arrow over (β)}_(j)=ω_(j) ²{right arrowover (β)}_(j)  (24)

[0129] The eigenvector of this equation, {right arrow over (β)}_(j),contains weighting factors that describe the j^(th) mistuned mode as asum of tuned modes. The corresponding eigenvalue, ω_(j) ², is the j^(th)mode's natural frequency squared. The matrix of the eigenvalue problemcontains two terms, Ω° and {overscore (Ω)}. Ω° is a diagonal matrix ofthe tuned system frequencies, ordered by ascending inter-blade phaseangle of their corresponding mode. The notation a Ω°² is shorthand forΩ°^(T)Ω°, which results in a diagonal matrix of the tuned systemfrequencies squared. The matrix {overscore (Ω)} contains the discreteFourier transforms (DFT) of the sector frequency deviations.

[0130] As discussed earlier, FMM treats the rotor's mistuning as a knownquantity that it uses to determine the system's mistuned modes andfrequencies. However, if the mistuned modes and frequencies are treatedas known parameters, the inverse problem could be solved to determinethe rotor's mistuning. This is the basis of FMM ID methods.

[0131] The following describes manipulation of the FMM equation ofmotion to solve for the mistuning in the rotor. Thus, in equation (24),all quantities are treated as known except {overscore (Ω)}, whichdescribes the system's mistuning. Subtracting the Ω°² term from bothsides of equation (24) and regrouping terms yields

2Ω°{overscore (Ω)}[Ω°{right arrow over (β)}_(j)]=(ω_(j) ² I−Ω°²){rightarrow over (β)}_(j)  (25)

[0132] The bracketed quantity on the left-hand side of equation (25)contains a known vector, which may be denoted as {right arrow over(γ)}_(j),

{right arrow over (γ)}_(j)=Ω°{right arrow over (β)}_(j)  (26)

[0133] Thus, {right arrow over (γ)}_(j) contains the modal weightingfactors, {right arrow over (β)}_(j) scaled on an element-by-elementbasis by their corresponding natural frequencies. Substituting {rightarrow over (γ)}_(j) into equation (25) yields

2Ω°[{overscore (Ω)}{right arrow over (γ)}_(j)]=(ω_(j) ² I−Ω°²){rightarrow over (β)}_(j)  (27)

[0134] After some algebra, it can be shown that the product in thebracket in equation (27) may be rewritten in the form

{overscore (Ω)}{right arrow over (γ)}_(j)=Γ_(j){overscore (ω)}  (28)

[0135] where the vector {overscore (ω)} equals [{overscore (ω)}₀,{overscore (ω)}₁ . . . {overscore (ω)}_(N−1)]^(T). The matrix Γ_(j) iscomposed from the elements in {right arrow over (γ)}_(j) and has theform $\begin{matrix}{\Gamma_{j} = \begin{bmatrix}{\overset{\_}{\gamma}}_{j\quad 0} & {\overset{\_}{\gamma}}_{j1} & \cdots & {\overset{\_}{\gamma}}_{j\quad {({N - 1})}} \\{\overset{\_}{\gamma}}_{j1} & {\overset{\_}{\gamma}}_{j2} & \cdots & {\overset{\_}{\gamma}}_{j\quad 0} \\\vdots & \vdots & \quad & \vdots \\{\overset{\_}{\gamma}}_{j\quad {({N - 1})}} & {\overset{\_}{\gamma}}_{j\quad 0} & \cdots & {\overset{\_}{\gamma}}_{j\quad {({N - 2})}}\end{bmatrix}} & (29)\end{matrix}$

[0136] where γ_(jn) denotes the n^(th) element of the vector {rightarrow over (γ)}_(j); the {right arrow over (γ)}_(j) elements arenumbered from 0 to N−1. Note that each column of Γ_(j) is the negativepermutation of the previous column.

[0137] Substituting equation (28) into (27) produces an expression inwhich the matrix of mistuning parameters, {overscore (Ω)}, has beenreplaced by a vector of mistuning parameters, {overscore (ω)}.

2Ω°Γ_(j){overscore (ω)}=(ω_(j) ² I−Ω°²){right arrow over (β)}_(j)  (30)

[0138] Pre-multiplying equation (30) by (2Ω°Γ_(j))⁻¹ would solve thisexpression for the DFT (Discrete Fourier Transform) of the rotor'smistuning. Furthermore, the vector {overscore (ω)} can then be relatedto the physical sector mistuning through an inverse discrete Fouriertransform. However, equation (30) only contains data from one measuredmode and frequency. Therefore, error in the mode's measurement mayresult in significant error in the predicted mistuning. To minimize theeffects of measurement error, multiple mode measurements may beincorporated into the solution for the mistuning. Equation (30) may beconstructed for each of the {circumflex over (M)} measured modes, andthe modes may be combined into the single matrix expression,$\begin{matrix}{{\begin{bmatrix}{2\quad \Omega {{}_{}^{}{}_{}^{}}} \\{2\quad \Omega {{}_{}^{}{}_{}^{}}} \\\vdots \\{2\quad \Omega {{}_{}^{}{}_{}^{}}}\end{bmatrix}\overset{\overset{arrow}{\_}}{\omega}} = \begin{bmatrix}{( {{\omega_{1}^{2}I} - {\Omega \,^{\circ 2}}} ){\overset{arrow}{\beta}}_{1}} \\{( {{\omega_{2}^{2}I} - {\Omega \,^{\circ 2}}} ){\overset{arrow}{\beta}}_{2}} \\\vdots \\{( {{\omega_{m}^{2}I} - {\Omega \,^{\circ 2}}} ){\overset{arrow}{\beta}}_{m}}\end{bmatrix}} & (31)\end{matrix}$

[0139] For brevity, equation (31) may be rewritten as

{tilde over (L)}{overscore (ω)}={right arrow over ({tilde over(r)})}  (32)

[0140] where {tilde over (L)} is the matrix on the left-hand side of theexpression, and {right arrow over ({tilde over (r)})} is the vector onthe right-hand side. The “{tilde over ()}” is used to indicate thatthese quantities are composed by vertically stacking a set ofsub-matrices or vectors.

[0141] It is noted that the expression in equation (32) is anoverdetermined set of equations. Therefore, it may not be possible tosolve for a {overscore (ω)} by direct inverse. However, one can obtain aleast squares fit to the mistuning, i.e.

{overscore (ω)}=Lsq{{tilde over (L)}, {right arrow over ({tilde over(r)})}}  (33)

[0142] Equation (33) produces the vector {overscore (ω)} which best-fitsall the measured data. Therefore, the error in each measurement iscompensated for by the balance of the data. The vector {overscore (ω)}can then be related to the physical sector mistuning through the inversetransform, $\begin{matrix}{{\Delta \quad \omega_{\psi}^{(s)}} = {\sum\limits_{p = 0}^{N - 1}\quad {^{{- }\quad s\quad p\frac{2\quad \pi}{N}}{\overset{\_}{\omega}}_{p}}}} & (34)\end{matrix}$

[0143] where Δω_(ψ) ^((s)) is the sector frequency deviation of thes^(th) sector. The following section describes how equations (33) and(34) can be applied to determine a rotor's mistuning.

[0144] 2.1.2 Experimental Application of Basic FMM ID

[0145] To solve equation (33) and (34) for the sector mistuning, onemust first construct {tilde over (L)} and {right arrow over ({tilde over(r)})} from the tuned system frequencies and the mistuned modes andfrequencies. The tuned system frequencies can be calculated throughfinite element analysis of a tuned, cyclic symmetric, single blade/disksector model. However, the mistuned modes and frequencies must beobtained experimentally.

[0146] The modes used by basic FMM ID are circumferential modes,corresponding to the tip displacement of each blade on the rotor. Incase of isolated families of modes, it may be sufficient to measure thedisplacement of only one point per blade. In practice, modes andfrequencies may be obtained by first measuring a complete set offrequency response functions (FRFs). Then, the modes and frequencies maybe extracted from the FRFs using modal curve fitting software.

[0147] The mistuned frequencies obtained from the measurements appearexplicitly in the basic FMM ID equations as ω_(j). However, the mistunedmodes enter into the equations indirectly through the modal weightingfactors {right arrow over (β)}_(j). Each vector {right arrow over(β)}_(j) is obtained by taking the inverse discrete Fourier transform ofthe corresponding single point-per-blade mode, i.e., $\begin{matrix}{\beta_{j\quad n} = {\sum\limits_{m = 0}^{N - 1}{\varphi_{j\quad m}^{{- }\quad m\quad n\frac{2\quad \pi}{N}}}}} & (35)\end{matrix}$

[0148] The quantities obtained from equation (35) may then be used withthe tuned system frequencies to construct {tilde over (L)} and {rightarrow over ({tilde over (r)})} as outlined hereinbefore. Finally,equations (33) and (34) may be solved for the sector mistuning. Thisprocess is demonstrated through the two examples in the followingsection.

[0149] 2.1.3 Numerical Examples for Basic FMM ID

[0150] The first example considers an integrally bladed compressor whoseblades are geometrically mistuned. The sector frequency deviationsidentified by basic FMM ID are verified by comparing them with valuesdirectly determined by finite element analyses (FEA). The second examplehighlights basic FMM ID's ability to detect mistuning caused byvariations at the blade-disk interface. This example considers acompressor in which all the blades are identical, however they aremounted on the disk at slightly different stagger angles. The mistuningcaused by the stagger angle variations is determined by FMM ID andcompared with the input values.

[0151] 2.1.3.1 Geometric Blade Mistuning

[0152]FIG. 14 represents an exemplary finite element model 26 of atwenty blade compressor. Although the airfoils on this model are simplyflat plates, the rotor design reflects the key dynamic behaviors of amodern, integrally bladed compressor. The rotor was mistuned through acombination of geometric and material property changes. Approximatelyone-third of the blades were mistuned through length variations,one-third through thickness variations, and one-third through elasticmodulus variations. The magnitudes of the variations were chosen so thateach form of mistuning would contribute equally to a 1.5% standarddeviation in the sector frequencies.

[0153] A finite element analysis of the tuned rotor was first performedto generate its nodal diameter map. FIG. 15 illustrates the naturalfrequencies of the tuned compressor modeled in FIG. 14, i.e., the tunedrotor's nodal diameter map. It is observed from FIG. 15 that the lowestfrequency family of first bending modes was isolated (as denoted by therectangle portion 27) for the basic FMM ID analysis. The sectormistuning of this rotor was then determined through two differentmethods: finite element analyses (FEA) of the mistuned sectors using thecommercially available ANSYS finite element code, and basic FMM ID.

[0154] The finite element calculations serve as a benchmark to assessthe accuracy of the basic FMM ID method. In the benchmark, a finiteelement model was made for each mistuned blade. In the model the bladewas attached to a single disk sector. The frequency change in themistuned blade/disk sector was then calculated with various cyclicsymmetric boundary conditions applied to the disk. It was found that thephase angle of the cyclic symmetric constraint had little effect on thefrequency change caused by blade mistuning. The values described hereinwere for a disk phase constraint of 90 degrees, i.e., for the five nodaldiameter mode.

[0155] A finite element model of the full, mistuned bladed disk was alsoconstructed and used to compute its mistuned modes and naturalfrequencies. The modes and frequencies were used as input data for basicFMM ID. In another embodiment, the mistuned modes and frequencies may beobtained through a modal fit of the rotor's frequency responsefunctions. Typically, the measurements may not detect modes that have anode point at the excitation source. To reflect this phenomenon, allmistuned modes that had a small response at blade one were eliminated.This left 16 modes and natural frequencies to apply to basic FMM ID.

[0156] The mistuned modes and frequencies were combined with the tunedsystem frequencies of the fundamental mode family to construct the basicFMM ID equations (31). These equations were solved using a least squaresfit. The solution was then converted to the physical sector frequencydeviations through the inverse transform given in equation (34).

[0157]FIG. 16 shows the comparison between the sector mistuningcalculated directly by finite element simulations of each mistunedblade/sector and the mistuning identified by basic FMM ID. As is seenfrom FIG. 16, the two results are in good agreement.

[0158] 2.1.3.2 Stagger Angle Mistuning

[0159] One of the differences between basic FMM ID and other mistuningidentification methods is its measure of mistuning. Basic FMM ID uses afrequency quantity that characterized the mistuning of an entireblade-disk sector, whereas other methods in the literature considermistuning to be confined to the blades as can be seen, for example, fromthe discussion in Judge, J. A., Pierre, C., and Ceccio, S. L., 2002,“Mistuning Identification in Bladed Disks,” Proceedings of theInternational Conference on Structural Dynamics Modeling, MadeiraIsland, Portugal. The sector frequency approach used by FMM not onlyidentifies the mistuning in the blades, but it also captures themistuning in the disk and the blade-disk interface. To highlight thiscapability, the following example considers a rotor in which the bladesare identical except they are mounted on the disk with slightlydifferent stagger angles. FIG. 17 schematically illustrates a rotor 29with exaggerated stagger angle variations as viewed from above.

[0160] In case of the compressor 26 in FIG. 14, its rotor was mistunedby randomly altering the stagger angle of each blade with a maximumvariation of ±4°. Otherwise the blades were identical. The modes of thesystem were then calculated using the ANSYS® finite element code. FIG.18 shows a representative mistuned mode caused by stagger anglemistuning of the rotor in FIG. 14. It is seen in FIG. 18 that the modewas localized (in the higher blade number region), indicating thatvarying the stagger angles does indeed mistune the system.

[0161] The mistuned modes and frequencies calculated by ANSYS® softwarewere then used to perform a basic FMM ID analysis of the mistuning. Theresulting sector frequency deviations are plotted as the solid line inFIG. 19, which illustrates a comparison of mistuning determination frombasic FMM ID and the variations in the stagger angles. The circles inFIG. 19 correspond to the stagger angle variations applied to eachblade. The vertical axes in FIG. 19 have been scaled so that the maximumfrequency and angle variation data points (blade 14) are coincident.This was done to highlight the fact that the stagger angle variationsare proportional to the sector frequency deviations detected by basicFMM ID. Thus, not only can basic FMM ID substantially accurately detectmistuning in the blades, as illustrated in the previous example, but itcan also substantially accurately detect other forms of mistuning suchas variation in the blade stagger angle.

[0162] 2.2 Advanced FMM ID Method

[0163] As discussed before, the basic FMM ID method provides aneffective means of determining the mistuning in an IBR. The basic FMM IDtechnique requires a set of simple vibration measurements and thenatural frequencies of the tuned system. However, at times neither thetuned system frequencies nor a finite element model from which to obtainthem are available to determine an IBR's mistuning. Furthermore, even ifa finite element model is available, there is often concern as to howaccurately the model represents the actual rotor. Therefore, thefollowing describes an alternative FMM ID method (advanced FMM ID) thatdoes not require any analytical data. Advanced FMM ID requires only alimited number of mistuned modes and frequency measurements to determinea bladed disk's mistuning. Furthermore, the advanced FMM ID method alsoidentifies the bladed disk's tuned system frequencies. Thus, advancedFMM ID not only serves as a method of identifying mistuning of thesystem, but can also provide a method of corroborating the finiteelement model of the tuned system

[0164] 2.2.1 Advanced FMM ID Theory

[0165] Advanced FMM ID may be derived from the basic FMM ID equations.Recall that a step in the development of the basic FMM ID theory was totransform the mistuning matrix {overscore (Ω)} into a vector form. Oncethe mistuning was expressed as a vector, it could then be calculatedusing standard methods from linear algebra. A similar approach is usedbelow to solve for the tuned system frequencies. However, the resultingequations are nonlinear, and require a more sophisticated solutionapproach.

[0166] 2.2.1.1 Development of Nonlinear Equations

[0167] Consider the basic FMM ID equation (30). Moving the Ω°² term tothe left-hand side of the equation, the expression becomes

Ω°²{right arrow over (β)}_(j)+2Ω°Γ_(j){overscore (ω)}=ω_(j) ²{rightarrow over (β)}_(j)  (36)

[0168] It is assumed that from measurement of the mistuned modes andfrequencies, {right arrow over (β)}_(j) and ω_(j) in equation (36) areknown. All other quantities are unknown. It is noted that although Γ_(j)is not known, the matrix contains elements from {right arrow over(β)}_(j). Therefore, some knowledge of the matrix is available.

[0169] After some algebra, one can show that the term Ω°²{right arrowover (β)}_(j) in equation (36) may be re-expressed as

Ω°²{right arrow over (β)}_(j) =B _(j){right arrow over (λ)}°  (37)

[0170] where {right arrow over (λ)}° is a vector of the tunedfrequencies squared, and B_(j) is a matrix composed from the elements of{right arrow over (β)}_(j). If η is defined to be the maximum number ofnodal diameters on the rotor, i.e. η=N/2 if N is even or (N−1)/2 if N isodd, then {right arrow over (λ)}° is given by $\begin{matrix}{{\overset{arrow}{\lambda}}^{\circ} = \begin{bmatrix}\omega_{0{ND}}^{\circ 2} \\\omega_{1{ND}}^{\circ 2} \\\vdots \\\omega_{\eta ND}^{\circ 2}\end{bmatrix}} & (38)\end{matrix}$

[0171] For N even, the matrix B_(j) has the form $\begin{matrix}{B_{j} = \begin{bmatrix}\beta_{j0} & \quad & \quad & \quad & \quad \\\quad & \beta_{j1} & \quad & \quad & \quad \\\quad & \quad & \beta_{j2} & \quad & \quad \\\quad & \quad & \quad & ⋰ & \quad \\\quad & \quad & \quad & \quad & \beta_{j\quad \eta} \\\quad & \quad & \quad & \ddots & \quad \\\quad & \quad & \beta_{j2} & \quad & \quad \\\quad & \beta_{j1} & \quad & \quad & \quad\end{bmatrix}} & (39)\end{matrix}$

[0172] A similar expression can be derived for N odd.

[0173] Substituting equation (37) into (36) and regrouping the left-handside results in a matrix equation for the tuned frequencies squared andthe sector mistuning, $\begin{matrix}{{\begin{bmatrix}B_{j} & {2\quad \Omega^{\circ}\Gamma_{j}}\end{bmatrix}\begin{bmatrix}\overset{arrow}{\lambda} \\\overset{\overset{arrow}{\_}}{\omega}\end{bmatrix}} = {\omega_{j}^{2}{\overset{arrow}{\beta}}_{j}}} & (40)\end{matrix}$

[0174] Equation (40) contains information from only one of the Mmeasured modes and frequencies. However, equation (40) can beconstructed for each measured mode, and combined into the single matrixexpression $\begin{matrix}{{\begin{bmatrix}B_{1} & {2\quad \Omega^{\circ}\Gamma_{1}} \\B_{2} & {2\quad \Omega^{\circ}\Gamma_{2}} \\\vdots & \vdots \\B_{M} & {2\quad \Omega^{\circ}\Gamma_{M}}\end{bmatrix}\begin{bmatrix}\overset{arrow}{\lambda} \\\overset{\overset{arrow}{\_}}{\omega}\end{bmatrix}} = \begin{bmatrix}\begin{matrix}\begin{matrix}{\omega_{1}^{2}{\overset{arrow}{\beta}}_{1}} \\{\omega_{2}^{2}{\overset{arrow}{\beta}}_{2}}\end{matrix} \\\vdots\end{matrix} \\{\omega_{M}^{2}{\overset{arrow}{\beta}}_{M}}\end{bmatrix}} & (41)\end{matrix}$

[0175] Thus, equation (41) represents a single expression thatincorporates all of the measured data. For brevity, equation (41) isrewritten as $\begin{matrix}{{\begin{matrix}\lbrack \overset{\sim}{B}  & {2( \overset{\sim}{ {\Omega^{\circ}\Gamma} )} \rbrack}\end{matrix}\begin{bmatrix}\overset{arrow}{\lambda} \\\overset{\overset{arrow}{\_}}{\omega}\end{bmatrix}} = {\overset{\overset{\sim}{arrow}}{r}}^{\prime}} & (42)\end{matrix}$

[0176] where {tilde over (B)} is the stacked matrix of B_(j), the term({tilde over (Ω°Γ)}) is the stacked matrix of Ω°Γ_(j), and {right arrowover ({tilde over (r)})}′ is the right-hand side of equation (41).

[0177] An additional constraint equation may be required because theequations (42) are underdetermined. To understand the cause of thisindeterminacy, consider a rotor in which each sector is mistuned thesame amount. Due to the symmetry of the mistuning, the rotor's modeshapes will still look tuned, but its frequencies will be shifted. Ifone has no prior knowledge of the tuned system frequencies, there maynot be any way to determine that the rotor has in fact been mistuned.The same difficulty arises in solving equation (42) because there maynot be any way to distinguish between a mean shift in the mistuning anda corresponding shift in the tuned system frequencies. To eliminate thisambiguity, mistuning may be defined so that it has a mean value of zero.

[0178] Mathematically, a zero mean in the mistuning translates toprescribing the first element of {overscore (ω)} to be zero. With theaddition of this constraint, equation (42) takes the form$\begin{matrix}{{\begin{bmatrix}\overset{\sim}{B} & {2\quad ( \overset{\sim}{ {\Omega^{\circ}\Gamma} )} } \\0 & \overset{arrow}{c}\end{bmatrix}\begin{bmatrix}{\overset{arrow}{\lambda}}^{\circ} \\\overset{\overset{arrow}{\_}}{\omega}\end{bmatrix}} = \begin{bmatrix}\overset{\overset{\sim}{arrow}}{r} \\0\end{bmatrix}} & (43)\end{matrix}$

[0179] where {right arrow over (c)} is a row vector whose first elementis 1 and whose remaining elements are zero.

[0180] 2.2.1.2 Iterative Solution Method

[0181] If the term ({tilde over (Ω°Γ)}) in equation (43) were known,then a least squares solution could be obtained for the tunedeigenvalues {right arrow over (λ)}° and the DFT of the sector mistuning{overscore (ω)}. However, because ({tilde over (Ω°Γ)}) is based in parton the unknown quantities {right arrow over (λ)}°, the equations in (43)are nonlinear. Therefore, an alternative solution method may be devised.In a solution described below, an iterative approach is used to solvethe equations in (43).

[0182] In iterative form, the least squares solution to equation (43)can be written as $\begin{matrix}{\begin{bmatrix}{\overset{arrow}{\lambda}}^{\circ} \\\overset{\overset{arrow}{\_}}{\omega}\end{bmatrix}_{(k)} = {{Lsq}\{ {\begin{bmatrix}\overset{\sim}{B} & {2\quad ( {\overset{\sim}{ {\Omega^{\circ}\Gamma} )}}_{({k - 1})} } \\0 & \overset{arrow}{c}\end{bmatrix},\begin{bmatrix}\overset{\overset{\sim}{arrow}}{r} \\0\end{bmatrix}} \}}} & (44)\end{matrix}$

[0183] where the subscripts indicate the iteration number. For eachiteration, a new matrix ({tilde over (Ω°Γ)}) may be constructed based onthe previous iteration's solution for {right arrow over (λ)}°. Thisprocess may be repeated until a converged solution is obtained. With agood initial guess, this method may typically converge within a fewiterations.

[0184] To identify a good initial guess, in case of analyzing anisolated family of modes, it is observed that generally the frequenciesof isolated mode families tend to span a fairly small range. Therefore,one good initial guess is to take all of the tuned frequencies to beequal to one another, and assigned the value of the mean tunedfrequency, i.e.

{right arrow over (λ)}₍₀₎°=ω_(avg)°²  (45)

[0185] However, the value of ω_(avg)° is not known and therefore cannotbe directly applied to equation (44). Consequently, equation (43) may beslightly modified to incorporate the initial guess defined by equation(45). In equation (43), if the tuned frequencies are taken to be equalto ω_(avg)°, then the term ({tilde over (Ω°Γ)}) may be expressed as

({tilde over (Ω°Γ)})=ω_(avg)°{tilde over (Γ)}  (46)

[0186] where {tilde over (Γ)} is the matrix formed by verticallystacking the M Γ_(j) matrices.

[0187] The matrix Γ_(j) is also related to the tuned frequencies. As aresult, the elements of each matrix Γ_(j) simplify to the formω_(avg)°β_(jn). This allows one to rewrite Γ_(j) as

Γ_(j)=ω_(avg)°Z_(j)  (47)

[0188] where Z_(j) is composed of the elements β_(jn) in arranged in thesame pattern as the γ_(jn) elements shown in equation (29). Thus,consolidating all ω_(avg)° terms, equation (46) can be written as

({tilde over (Ω°Γ)})=ω_(avg)°²{tilde over (Z)}  (48)

[0189] where {tilde over (Z)} is the stacked form of the Z_(j) matrices.

[0190] Substituting equation (48) into equation (43) and regroupingterms results in the expression $\begin{matrix}{{\begin{bmatrix}\overset{\sim}{B} & {2\quad \overset{\sim}{Z}} \\0 & \overset{arrow}{c}\end{bmatrix}\begin{bmatrix}{\overset{arrow}{\lambda}}^{\circ} \\{\omega_{avg}^{\circ 2}\overset{\overset{arrow}{\_}}{\omega}}\end{bmatrix}} = \begin{bmatrix}\overset{\overset{\sim}{arrow}}{r} \\0\end{bmatrix}} & (49)\end{matrix}$

[0191] Note that the ω_(avg)°² term was grouped with the vector{overscore (ω)}. Thus, all the unknown expressions are consolidated intothe single vector on the left-hand side of equation (49). Thesequantities can be solved through a least squared fit of the equations.This represents the 0^(th) iteration of the solution process. The {rightarrow over (λ)}° terms of the solution may then be used as an initialguess for the first iteration of equation (44).

[0192] In practice, the mistuned modes and frequencies may be measuredusing the technique described for basic FMM ID in Section 2.1.3.2. Thenext section presents a numerical example that demonstrates the abilityof the advanced FMM ID method to identify the frequencies of the tunedsystem as well as mistuned sector frequencies.

[0193] 2.2.2 Numerical Test Case for Advanced FMM ID

[0194] This section presents a numerical example of the advanced FMM IDmethod that identifies the tuned system frequencies as well as themistuning. This example uses the same geometrically mistuned compressormodel 26 (FIG. 14) as that used for the basic FMM ID method. The tunedsystem frequencies and sector mistuning identified by advanced FMM IDare then compared with finite element results.

[0195] The modes and natural frequencies of the mistuned bladed diskwere calculated using a finite element model of the mistuned system. Thephysical modes were then converted to vectors of modal weightingfactors, {right arrow over (β)}_(j), through equation (35). Theweighting factors were used to form the elements of equation (49) whichwas solved to obtain an initial estimate of the tuned systemfrequencies. This initial estimate was used as an initial guess toiteratively solve equation (44). The solution vector contained twoparts: a vector of the tuned system frequencies squared, and a vector ofthe DFT of the sector frequency deviations. The sector mistuning wasconverted to the physical domain using the inverse transform in equation(34).

[0196] The resulting sector frequency deviations were compared with thebenchmark finite element analysis (FEA) values. FIG. 20 depicts acomparison of mistuning predicted using advanced FMM ID with thatobtained using the finite element analysis (FEA). The results in FIG. 20were obtained using the same procedure as that discussed in section2.1.3 above. FIG. 21 shows a comparison of the tuned frequenciesidentified by advanced FMM ID and those computed directly with thefinite element model (i.e., FEA). In each of FIGS. 20 and 21, theresults obtained using advanced FMM ID were in good agreement with thosefrom FEA.

[0197] [3] System Identification: Application

[0198]FIG. 22 illustrates an exemplary setup 32 to measure transferfunctions of test rotors and also to verify various FMM ID methodsdiscussed hereinbefore. As discussed earlier, the advanced FMM ID methoduses the measurements of the mistuned rotor's system modes and naturalfrequencies. The term “system mode” is used herein to refer to the tipdisplacement of each blade as a function of blade's angular position.The system modes may be obtained using a standard modal analysisapproach: measure the bladed disk's transfer functions, and thencuve-fit the transfer functions to obtain the mistuned modes and naturalfrequencies. The setup 32 in FIG. 22 may be used to perform standardtransfer function measurements. As illustrated in FIG. 22, the rotor tobe tested (rotor 34) may be placed on a foam pad 36 to approximate afree boundary condition. Then, one of the rotor blades may be excitedover the frequency range of interest using an excitation source 38 (forexample, a function generator coupled to an audio amplifier andloudspeaker) and the response of each blade may be measured with a laservibrometer 40 coupled to a spectrum analyzer 42, which can be used toanalyze the output of the laser vibrometer 40 to determine the transferfunction. The devices 38, 40, and 42 may be obtained from anycommercially available sources as is known in the art. For example, thecompanies that make the function generator and spectrum analyzer includeHewlett-Packard, Agilent, and Tektronix. The laser vibrometer may be aPolytec or Ometron vibrometer.

[0199] All of the devices 38, 40, 42 used in the test setup 32 are showncoupled (directly or indirectly through another device) to a computer44, which may be used to operate the devices as well as to analyzevarious data received from the devices. The computer 44 may also storethe FMM software 46, which can include software to implement any or allof the FMM ID methods. It is understood by one skilled in the art thatthe FMM software module 46 may be stored on an external magnetic,electromagnetic or optical data storage medium (not shown) such as, forexample, a compact disc, an optical disk, a floppy diskette, etc. Thedata storage medium may then be supplied to the appropriate reader unitin the computer 44 or attached to the computer 44 to read the content ofthe data storage medium and supply the FMM software to the computer 44for execution. Alternatively, the FMM software module 46 may reside inthe computer's internal memory such as, for example, a hard disk drive(HDD) from which it can be executed by the computer's operating system.It is apparent to one skilled in the art that the computer 44 may be anycomputing unit including, for example, a stand-alone or networked IBM-PCcompatible computer, a computing work station, etc.

[0200] It is noted here that for the sake of convenience and brevity thefollowing discussion uses the term “FMM ID” to refer to any of the basicas well as the advanced FMM ID methods without specifically identifyingeach one. However, based on the context of the discussion and thediscussion presented hereinbefore, it would not be difficult for oneskilled in the art to comprehend which one of the two FMM ID methods isbeing referred to in the discussion.

[0201] To investigate applicability of FMM ID methods to realexperimental data from actual hardware, the methods were applied to apair of transonic compressors whose corresponding test rotors weredesignated as SN-1 and SN-3. A single blade/disk sector finite elementmodel of the tuned compressor was provided by Pratt and Whitney. Bysolving this model with free boundary conditions at the hub and variouscyclic symmetric boundary conditions on the radial boundaries of thedisk, a nodal diameter map of the tuned rotor was generated asillustrated in FIG. 23. The free boundary conditions at the hubrepresented the boundary conditions in the experiment: an IBR supportedby a soft foam pad and is otherwise unconstrained. In FIG. 23, each ofthe first two families of modes (designated by reference numerals 50 and52) have isolated frequencies. These correspond to first bending andfirst torsion modes, respectively. Because FMM ID is equally applicablefor isolated families of modes, both the first bending and first torsionmodes were suitable candidates for system identification analysis.

[0202]FIG. 24 illustrates a typical transfer function from compressorSN-1 obtained using the test setup 32 shown in FIG. 22. Note that due tothe high modal density, it was necessary to measure the compressorfrequency response with a very high frequency resolution. This processwas repeated for both compressors over two frequency bands to capturethe response of both the first bending and first torsion modes. Thecommercially available MODENT modal analysis package was then used tocurve-fit the transfer functions. This resulted in measurements of themistuned first bending and torsion modes of each rotor, along with theirnatural frequencies. Because the blade that was excited was at a lowresponse point in some modes, two or three of the modes in each familywere not measurable. In any event, the measured mistuned modes andnatural frequencies were used to demonstrate the applicability of FMM IDto actual hardware.

[0203] 3.1 FMM ID Results

[0204] The measured modes and frequencies were used to test both formsof the FMM ID method. The basic and advanced FMM ID methods were appliedto each rotor, for both the first bending and torsion families of modes.The tuned frequencies required by basic FMM ID were the same as thosedepicted in FIG. 23. To assess the accuracy of FMM ID, the results werecompared to benchmark data.

[0205] 3.1.1 Benchmark Measure of Mistuning

[0206] To assess the accuracy of the FMM ID method, the results must becompared to benchmark data. However, because the test rotors wereintegrally bladed, their mistuning could not be measured directly.Therefore, an indirect approach was used to obtain the benchmarkmistuning. Pratt and Whitney personnel carefully measured the geometryof each blade on the two rotors and calculated the frequencies that itwould have if it were clamped at its root. Because each blade had aslightly different geometry, it also had slightly different frequencies.Thus, the variations in the blade frequencies caused by geometricvariations were determined. This data was put in a form that could becompared with the values identified by FMM ID. First, the frequencyvariations as a fraction of the mean were calculated so that thedeviation in the blade frequencies could be determined. These in turnwere related to the sector frequency deviations determined by FMM ID.For modes with most of their strain energy in the blade, sectorfrequency deviations can be obtained from blade frequency mistuning bysimple scaling, i.e.

Δω_(ψ)=a(Δω_(b))  (50)

[0207] where a is the fraction of strain energy in the blade for theaverage nodal diameter mode.

[0208] 3.1.2 FMM ID Results for Bending Modes

[0209] SN-1 Results

[0210] The measured mistuned modes and natural frequencies for thecompressor SN-1 were used as input to both versions of FMM ID. In thecase of basic FMM ID, the tuned system frequencies of the first bendingfamily from FIG. 23 were also used as input. FIG. 25 illustrates acomparison of mistuning from each FMM ID method with benchmark resultsfor a test rotor SN-1. FIG. 25 thus shows the sector frequencydeviations identified by each FMM ID method along with the benchmarkresults. Both FMM ID methods were in good agreement with the benchmark.This may imply that the mistuning was predominantly caused by geometricvariations and that the variations were accurately captured by Pratt andWhitney.

[0211] To make the comparisons easier, all mistuning in FIG. 25 wasplotted as the variation from a zero mean. However, it is noted thatrotor SN-1 had a mean frequency 1.3% higher than that of the tunedfinite element model. This DC shift was detected by basic FMM ID as aconstant amount of mistuning added to each blade's frequency. However,because the advanced FMM ID formulation does not incorporate the tunedfinite element frequencies, it had no way to distinguish between a meanshift in the mistuning and a corresponding shift in the tuned systemfrequencies. Therefore, in advanced FMM ID, mistuning was defined tohave a zero mean, and then a corresponding set of tuned frequencies wasinferred.

[0212]FIG. 26 shows a comparison of tuned system frequencies for thetest rotor SN-1 from advanced FMM ID (i.e., identified by advanced FMMID) and the finite element model (FMM) using ANSYS® software. It is seenfrom FIG. 26 that the FMM ID frequencies were approximately 17 Hz higherthan the finite element values. This corresponds to a 1.3% shift in themean of the tuned system frequencies that compensated for fact that theblade mistuning now had a zero mean. To facilitate the comparison of thefinite element and FMM ID results, the mean shift was subtracted andthen the results were then plotted as circles on FIG. 26. After thisadjustment, it is seen that the distribution of the tuned frequenciesdetermined by FMM ID agreed quite well with the values calculated fromthe finite element model. Advanced FMM ID additionally identified thefact that SN-1 had slightly higher average frequencies than the FEMmodel—a fact that could be important in establishing frequency marginsfor the stage.

[0213] It is observed from the sector frequency deviations of SN-1 shownin FIG. 25 that the mistuning varied from blade-to-blade in a regularpattern. The decreasing pattern of mistuning and the jump in the patternmay suggest that the mistuning might have been caused by tool wearduring the machining process and that an adjustment in the process wasmade during blade manufacturing.

[0214] SN-3 Results

[0215] The basic and advanced FMM ID methods were then applied in asimilar manner to rotor SN-3's family of first bending modes. Theidentified mistuning and tuned system frequencies are shown in FIGS. 27and 28, respectively. For comparison purposes, the mistuning was againplotted with a zero mean, and a corresponding mean shift was subtractedfrom the predicted tuned system frequencies. The predictions for rotorSN-3 from both FMM ID methods were also in good agreement with thebenchmark results. It is noted that in FIG. 27, the blades were numberedso that blade-1 corresponded to the high frequency sector. A similarnumbering scheme (not illustrated here) was also implemented for SN-1for comparison.

[0216] 3.1.3 FMM ID Results for Torsion Modes

[0217] In this section, FMM ID's ability to identify mistuning in thefirst torsion modes is examined. For brevity, only the results foradvanced FMM ID are presented. Advanced FMM ID was applied to each testrotor's family of torsion modes. FIGS. 29(a) and (b) show a comparison,for rotors SN-1 and SN-3 respectively, of the mistuning identified byFMM ID with the values from benchmark results obtained by Pratt &Whitney from geometric measurements. The agreement between FMM ID andbenchmark results is good. In FIG. 29, the blades were numbered in thesame order as in FIG. 27, which represents the numbering for SN-3 but,although not shown, a similar numbering for SN-1 was also employed.Thus, the mistuning patterns in the torsion modes looked very similar tothose observed for the bending modes, e.g., the blades with the highestand lowest frequencies were the same for both sets of modes. Thissuggests that the mistuning in SN-1 and SN-3 systems might have beencaused by relatively uniform thickness variations in the blades becausesuch mistuning would affect the frequencies of both types of modes in avery similar manner.

[0218] In addition to identifying the mistuning in these rotors,advanced FMM ID also simultaneously inferred the tuned systemfrequencies of the system's torsion modes, as shown in FIG. 30, whichillustrates a comparison of tuned system frequencies from advanced FMMID and ANSYS® software for torsion modes of rotors SN-1 and SN-3. Thus,FMM ID worked well on both the torsion and bending modes of the testcompressors.

[0219] 3.2 Forced Response Prediction

[0220] The mistuning identified in section 3.1 was used to predict theforced response of the test compressors (SN-1, SN-3) to a traveling waveexcitation. The results were compared with benchmark measurements doneby Pratt & Whitney.

[0221] Pratt and Whitney has developed an experimental capability forsimulating traveling wave excitation in stationary rotors. Theirtechnique was applied to SN-1 to measure its first bending family'sresponse to a 3E excitation (third engine order excitation). Theresponse of SN-1 was then predicted using FMM ID methods. To make theprediction, the mistuning and tuned system frequencies identified byadvanced FMM ID (as discussed in section 3.1) were input to the FMMreduced order model discussed hereinabove under part [1]. FMM calculatedthe system's mistuned modes and natural frequencies. Then, modalsummation was used to calculate the response to a 3E excitation. Themodal damping used in the summation was calculated from the half-powerbandwidth of the transfer function peaks.

[0222]FIG. 31(a) depicts FMM-based forced response data, whereas FIG.31(b) depicts the experimental forced response data. Thus, the plots inFIG. 31 show the comparison of the benchmark forced response resultswith that predicted by FMM. For clarity, only the envelope of the bladeresponse is shown in FIGS. 31(a), (b). Also, the plots in FIG. 31 havebeen normalized so that the maximum response is equal to one. Ingeneral, the two curves in FIG. 31 agree reasonably well. To observe howwell the response of individual blades was predicted, the relativeresponses of the blades at two resonant peaks were compared. The peaksare labeled {circumflex over (1)} and {circumflex over (2)} in FIG.31(a). FIGS. 32(a) and (b) respectively show relative blade amplitudesat forced response resonance for the resonant peaks labeled {circumflexover (1)} and {circumflex over (2)} in FIG. 3 1(a). The relativeamplitude of each blade as determined by FMM and experimental methods isplotted for both resonant peaks in FIG. 32. The agreement between FMMand experimental predictions was reasonably good. Thus, the FMM basedmethod may be used to not only capture the overall shape of theresponse, but also to determine the relative amplitudes of the blades atthe various resonances.

[0223] 3.3 Cause and Implications of Repeated Mistuning Pattern

[0224] The mistuning in bladed disks is generally considered to be arandom phenomenon. However, it is seen from the discussion in section3.1 that both test rotors SN-1 and SN-2 had very similar mistuningpatterns that were far from random. If such repeated mistuning mattersare found to be common among IBRs, it may have broad implications on thepredictability of these systems.

[0225] 3.3.1 Cause of Repeated Mistuning

[0226] The similarity between the mistuning patterns identified in SN-1and SN-3 is highly suggestive that the mistuning was caused by aconsistent manufacturing effect. In addition, it was observed that themistuning in the torsion modes followed the same trends as in thebending modes. Thus, the dominant form of mistuning may most likely becaused by relatively uniform blade-to-blade thickness variations. Bladethickness variations may be analyzed using geometry measurements of arotor to extract the thickness of each blade at different points acrossthe airfoil. Then, a calculation may be performed to determine how mucheach point's thickness deviated from the average values of allcorresponding points. The results can be expressed as a percentagevariation from the mean blade thickness. It was found that a 2% changein blade thickness, produced about a 1% change in corresponding sectorfrequency, which is consistent with beam theory for a beam of curvedcross-section.

[0227] It is observed that tool wear may cause blade thicknessvariations. For example, if the blades were machined in descending orderfrom blade 18 to blade 1 (e.g., the 18 blades in rotor SN-1), then, dueto tool wear, each subsequent blade would be slightly larger than theprevious one. This effect would cause the sector frequencies tomonotonically increase around the wheel. Any frequency jump ordiscontinuity observed (e.g., the jump at blade 15 in FIG. 25) may bethe result of a tool adjustment made during the machining process.

[0228] 3.3.2 Implications of Repeated Mistuning

[0229] The repeating mistuning patterns caused by machining effects mayallow prediction of the response of a fleet (e.g., of compressors)through probabilistic methods. For example, consider an entire fleet ofthe transonic compressors, two of which—SN-1 and SN-2—were discussedhereinbefore. If it is incorrectly assumed that the mistuning in theserotors was completely random, then one would estimate that the sectorfrequency deviation of each sector has a mean of zero and a standarddeviation of about 2%. Assuming these variations, FMM was used toperform 10,000 Monte Carlo simulations to represent how a fleet ofengines would respond to a 3E excitation. The data from Monte Carlosimulations was used to compute the cumulative probability function(CPF) of the maximum blade amplitude on each compressor in the fleet.FIG. 33 depicts cumulative probability function plots of peak bladeamplitude for a nominally tuned and nominally mistuned compressor. TheCPF of a fleet of engines with random mistuning had a standard deviationof 2% as shown by the dashed line in FIG. 33. It is observed from FIG.33 that the maximum amplitude varied widely across the fleet, ranging inmagnification from 1.1 to 2.5.

[0230] However, the test rotors were in fact nominally mistuned with asmall random variation about the nominal pattern. Because the randomvariation was much smaller than that considered above, the fleet'sresponse was more predictable. To illustrate this point, the nominalmistuning pattern (of the fleet of rotors) was approximated as the meanof the patterns measured for the two test rotors SN-1 and SN-2. Based onthis approximated pattern, it was found that the sector frequencydeviations differed from the nominal values with a standard deviation ofonly 0.2%, as shown in FIG. 34, which shows mean and standard deviationsof each sector's mistuning for a nominally mistuned compressor. Makinguse of the fact that the rotors were nominally mistuned, the Monte Carlosimulations were repeated. The CPF of the maximum amplitude on eachrotor was then calculated. The calculated results were plotted as thesolid line on FIG. 33. It is observed that by accounting for nominalmistuning, the range of maximum amplitudes is significantly reduced.Thus, by measuring and making use of nominal mistuning when it occurs, atest engineer may predictably determine the fleet's vibratory responsebehavior from the vibratory response of a specific IBR that is tested ina spin pit, rig test or engine.

[0231] [4] Mistuning Extrapolation for Rotation

[0232] The FMM ID methods presented earlier in part [3] determine themistuning in a bladed disk while it is stationary. However, once therotor is spinning, centrifugal forces can alter its effective mistuning.However, an analytical method, discussed below, may be used forapproximating the effect of rotation speed on mistuning.

[0233] 4.1 Mistuning Extrapolation Theory

[0234] Centrifugal effects cause the sector frequency deviations presentunder rotating conditions to differ from their values when the bladeddisk is not rotating. To approximate the effect of rotational speed onmistuning, a lumped parameter model 54 of a rotating blade, as shown inFIG. 35, may be considered. The pendulum 56 mounted on a torsion spring58 represents the blade, while the circular region 60 of the systemrepresents a rigid disk. Thus, the blade is modeled as a pendulum 56 ofmass “m” and length “I” which is mounted to a rigid disk 60 through atorsional spring “k” 58. The disk 60 has radius “L” and rotates at speed“S”.

[0235] It can be shown that the blade's natural frequency in this systemis given by the expression $\begin{matrix}{{\omega (S)}^{2} = {\frac{k}{m\quad l^{2}} + {\frac{L}{l}S^{2}}}} & (51)\end{matrix}$

[0236] where S is the rotation speed in radians/sec, and the notationω(S) indicates the natural frequency at speed S. Notice that thequantity k/ml² is the natural frequency of the system at rest.Therefore, equation (51) can be rewritten in the more general form

ω(S)²=ω(0)² +rS ²  (52)

[0237] where r is a constant.

[0238] Take ω to be a mistuned frequency in the formω(S)=ω°(S)[1+Δω(S)]. Substituting this expression into equation (52) andkeeping only the first order terms implies

ω(S)²≈ω°(S)²+2(Δω(0)ω°(0)²  (53)

[0239] where ω°(S) is the tuned frequency at speed.

[0240] Taking the square root of expression (53) and again keeping onlythe first order terms one obtains an expression for the mistunedfrequency at speed, $\begin{matrix}{{\omega (S)} \approx {{\omega^{\circ}(S)}\{ {1 + {\Delta \quad {{\omega (0)}\lbrack \frac{{\omega^{\circ}(0)}^{2}}{{\omega^{\circ}(S)}^{2}} \rbrack}}} \}}} & (54)\end{matrix}$

[0241] Subtracting and dividing both sides of the expression (54) byω°(S) yields an approximation for the mistuned frequency ratio at speed,i.e. $\begin{matrix}{{\Delta \quad {\omega (S)}} \approx {\Delta \quad {{\omega (0)}\lbrack \frac{{\omega^{\circ}(0)}^{2}}{{\omega^{\circ}(S)}^{2}} \rbrack}}} & (55)\end{matrix}$

[0242] In the case of system modes in which the strain energy isprimarily in the blades, the tuned system frequencies tend to increasewith speed by the same percentage as the blade alone frequencies.Therefore, expression (55) can also be approximated by noting how afrequency of the tuned system changes with speed, e.g., $\begin{matrix}{{\Delta \quad {\omega (S)}} \approx {\Delta \quad {{\omega (0)}\lbrack \frac{{\omega_{\psi}^{\circ}(0)}^{2}}{{\omega_{\psi}^{\circ}(S)}^{2}} \rbrack}}} & (56)\end{matrix}$

[0243] where ω_(ψ)° is the average tuned system frequency. Expression(56) may then be used to adjust the sector frequency deviations measuredat rest for use under rotating conditions.

[0244] 4.2 Numerical Test Cases

[0245] This section presents two numerical tests of the mistuningextrapolation theory. The first example uses finite element analysis ofthe compressor SN-1 discussed hereinbefore (see, for example, FIG. 22)to assess the accuracy of expression (56). Then, the second exampledemonstrates that this result may be combined with FMM ID and the FMMforced response software to predict the response of a rotor at speed.

[0246] 4.2.1 Compressor SN-1

[0247] As mentioned earlier, Pratt & Whitney personnel made carefulmeasurements of each blade's geometry and used this data to constructaccurate finite element models of all 18 airfoils in SN-1. Thus, thesefinite element models captured the small geometric variations from oneblade to the next.

[0248] Two of the airfoil models were randomly selected for use in thistest case. For the purpose of this study, the first airfoil representedthe tuned blade geometry, and the second represented a mistuned blade.Then, both blades were clamped at their root, and their naturalfrequencies were calculated using finite element analysis (FEA) softwareANSYS® . The values were obtained for the first three modescorresponding to first bending, first torsion, and second bendingrespectively. The calculations were then repeated with the addition ofrotational velocity loads to simulate centrifugal effects. Through thisapproach, the natural frequencies of both blades were obtained at fiverotation speeds ranging from 0 to 20,000 RPM.

[0249] Next, the frequency deviation of the mistuned blade wascalculated by subtracting the tuned frequencies from the mistunedvalues, and then dividing each result by its corresponding tunedfrequency. FIG. 36 shows a comparison of mistuning values analyticallyextrapolated to speed with an FEA (finite element analysis) benchmark.The results plotted as lines in FIG. 36 represent benchmark values onwhich to assess the accuracy of the analytical mistuning extrapolationmethod. Using expression (55), the frequency deviations calculated forthe stationary rotor were extrapolated to the same rotational conditionsconsidered in the benchmark calculation. The extrapolated results areshown as circles on FIG. 36. The agreement between extrapolated resultsand the results using the FEA benchmark were good for all three modes.Thus, expression (55) may be used to analytically extrapolate bladefrequency deviations to rotating conditions. It is noted that expression(55) was used here rather than expression (56) because the calculatednatural frequencies represented an isolated blade and not a blade/disksector. However, for the cases where FMM is applicable, a blade-alonefrequency differs from an average sector frequency by a multiplicativeconstant. Thus, expression (56) may also be suitable for mistuningextrapolation.

[0250] 4.2.2 Response Prediction at Speed

[0251] This section uses a numerical test case that shows how FMM ID,expression (56), and the FMM forced response software can be combined topredict the response of a bladed disk under rotating conditions.

[0252] The geometrically mistuned rotor illustrated in FIG. 14 had a6^(th) engine order crossing with the first bending modes at arotational speed of 20,000 RPM. However, to create a more severe testcase, it is assumed that the crossing occurred at 40,000 RPM.

[0253] To use FMM to predict the rotor's forced response at this speed(40,000 RPM), the FMM prediction software must be provided with thebladed disk's tuned system frequencies and the sector frequencydeviations that are present at 40,000 RPM. As part of the discussion insection (2.1.3.1) above, these two sets of parameters were determinedfor at-rest condition using ANSYS and basic FMM ID respectively.However, because both of these properties change with rotation speed,they must first be adjusted to reflect their values at 40,000 RPM.

[0254] To adjust the tuned system frequencies for higher rotationalspeed, tuned system frequencies were recalculated in ANSYS® softwareusing the centrifugal load option to simulate rotational effects. FIG.37 illustrates the effect of centrifugal stiffening on tuned systemfrequencies. As shown in FIG. 37, the centrifugal stiffening caused thetuned system frequencies to increase by about 30%. Then, the change inthe five nodal diameter, tuned system frequency and expression (56) wereused to analytically extrapolate the sector frequency deviations to40,000 RPM. The adjusted mistuning, along with the original mistuningvalues identified at rest, are plotted in FIG. 38, which illustrates theeffect of centrifugal stiffening on mistuning. It is seen from FIG. 38that the centrifugal loading reduces the mistuning ratios by 40%.

[0255] The adjusted parameters were then used with the FMM forcedresponse software to calculate the rotor's response to a 6E excitationusing the method described hereinabove in parts [1] and [2]. As abenchmark, the forced response was also calculated directly in ANSYS®software using a full 360° mistuned finite element model. Tracking plotsof the FMM and ANSYS® software results are shown in FIG. 39, whichdepicts frequency response of blades to a six engine order excitation at40,000 RPM rotational speed. For clarity, the response of only threeblades is shown in FIG. 39: the high responding blade, the medianresponding blade, and the low responding blade. It is observed from FIG.39 that each blade's peak amplitude and the shape of its overallresponse as predicted by FMM agree well with the benchmark results.Thus, FMM ID, the mistuning extrapolation equation, and FMM may becombined to identify the mistuning of a rotor at rest, and use themistuning to predict the system's forced response under rotatingconditions.

[0256] [5] System Identification from Traveling Wave ResponseMeasurements

[0257] Traditionally, mistuning in rotors with attachable blades ismeasured by mounting each blade in a broach block and measuring itsnatural frequency. The difference of each blade's frequency from themean value is then taken as a measure of its mistuning. However, thismethod cannot be applied to integrally bladed rotors (IBRs) whose bladescannot be removed for individual testing. In contrast, FMM ID systemidentification techniques rely on measurements of the bladed disk systemas a whole, and are thus well suited to IBRs.

[0258] FMM ID may also be used for determining the mistuning inconventional bladed disks. Even when applied to bladed disks withconventionally attached blades, the traditional broach block method ofmistuning identification is limited. In particular, it does not takeinto account the fact that the mistuning measured in the broach blockmay be significantly different from the mistuning that occurs when theblades are mounted on the disk. This variation can arise because eachblade's frequency is dependent on the contact conditions at theattachment. In the engine, the attachment is loaded by centrifugal forcefrom the blade which provides a different contact condition than theclamping action used in broach block tests. This difference isaccentuated in multi-tooth attachments because different teeth may comein contact depending on how the attachment load is applied. In addition,the contact in multi-tooth attachments may be sensitive to manufacturingvariations and, consequently, vary from one location to the next on thedisk. The discussion given below addresses these issues by devising amethod of system identification that can be used to directly determinemistuning while the stage is rotating, and can also identify mistuningfrom the response of the entire system because the blades are inherentlycoupled under rotating conditions. The method discussed below providesan approach for extracting the mistuned modes and natural frequencies ofthe bladed disk under rotating conditions from its response to naturallyoccurring, engine order excitations. The method is a coordinatetransformation that makes traveling wave response data compatible withthe existing, proven modal analysis algorithms. Once the mistuned modesand natural frequencies are known, they can be used as input to FMM IDmethods.

[0259] 5.1 Theory

[0260] Both of the FMM ID mistuning identification methods require themistuned modes and natural frequencies of the bladed disk as input.Under stationary conditions, they can be determined by measuring thetransfer functions of the system and using standard modal analysisprocedures. One way of measuring the transfer functions is to excite asingle point (e.g., on a blade) with a known excitation and measure thefrequency response of all of the other points that define the system.However, when the bladed disk is subjected to an engine order excitationall of the blades are simultaneously excited and it may not be clear howthe resulting vibratory response can be related to the transferfunctions typically used for modal identification. As discussed below,if the blade frequency response data is transformed in a particularmanner then the traveling wave excitation constitutes a point excitationin the transform space and that standard modal analysis techniques canthen be used to extract the transformed modes. Once the transformedmodes are determined, the physical modes of the system can be calculatedfrom an inverse transformation.

[0261] 5.1.1 Traditional Modal Analysis

[0262] Standard modal analysis techniques are based on measurements of astructure's frequency response functions (FRFs). These frequencyresponse functions are then assembled as a frequency dependant matrix,H(ω), in which the element H_(i,j)(ω) corresponds to the response ofpoint i to the excitation of point j as discussed, for example, inEwins, D. J., 2000, Modal Testing: Theory, Practice, and Application,Research Studies Press Ltd., Badlock, UK, Chapter 1. Traditional modalanalysis methods require that one row or column of this frequencyresponse matrix be measured. In the test cases discussed hereinbelow themistuned modes correspond to a single isolated family of modes. Forexample, the lower frequency modes such as first bending and firsttorsion families often have frequencies that are relatively isolated.When this is the case the “modes” of interest may be defined in terms ofhow the blade displacements vary from one blade to the next around thewheel and can be characterized by the response of one point per blade.Thus, the standard modal analysis experiment may be performed in one oftwo ways when measuring the mistuned modes of a bladed disk. First, thestructure's frequency response may be measured at one point on eachblade, while it is excited at only one blade. This would result in themeasurement of a single column of H(ω). Alternatively, a row of H(ω) maybe obtained by measuring the structure's response at only one blade andexciting the system at each blade in turn. In either of these acceptabletest configurations, the structure is excited at only one point at atime. However, in a traveling wave excitation, all blades are excitedsimultaneously. Thus, the response of systems subjected to suchmulti-point excitations cannot be directly analyzed by standard SISO(single input, single output) modal analysis methods.

[0263] 5.1.2 General Multi-Point Excitation Analysis

[0264] As discussed above, a traveling wave excitation is not directlycompatible with standard SISO modal analysis methods. Further, atraveling wave excites each measurement point with the same frequency atany given time. The method discussed below may be applicable to any.multi-input system, in which the frequency profile is consistent fromone excitation point to the next; however, the amplitude and phase ofthe excitation sources may freely vary spatially. It is noted thatsuitable excitation forms include traveling waves, acoustic pressurefields, and even shakers when appropriately driven.

[0265] In typical applications, the i,j element of the frequencyresponse matrix H(ω) corresponds to the response of point i to theexcitation of point j. However, to analyze frequency response data froma multi-point excitation, H_(i,j) (ω) may be viewed in a more generalfashion. Thus, in a more general sense, the i,j element describes theresponse of the i^(th) coordinate to an excitation at the j^(th)coordinate. Although these coordinates are typically taken to be thedisplacement at an individual measurement point, this need not be thecase.

[0266] The structure's excitation and response can instead betransformed into a different coordinate system. For example, an Ndegree-of-freedom coordinate system can be defined by a set of Northogonal basis vectors which span the space. In this representation,each basis vector is a coordinate. Thus, to perform modal analysis onmulti-point excitation data, it may be desirable to select a coordinatesystem in which the excitation is described by just one basis vector.Within this newly defined modal analysis coordinate system, thestructure is subjected to only a single coordinate excitation.Therefore, when the response measurements are expressed in this samedomain, they represent a single column of the FRF matrix, and can beanalyzed by standard SISO modal analysis techniques. The followingsection describes how this approach may be applied to traveling waveexcitations.

[0267] 5.1.3 Traveling Wave Modal Analysis

[0268] Consider an N-bladed disk subjected to a traveling waveexcitation. It is assumed that the amplitude and phase of each blade'sresponse is measured as a function of excitation frequency. In practice,these measurements may be made under rotating conditions with aNon-intrusive Stress Measurement System (NSMS), whereas a laservibrometer may be used in a stationary bench test. For simplicity, onlyconsider one measurement point per blade is considered.

[0269] It is assumed that the blades are excited harmonically by theforce {right arrow over (f)}(ω)e^(iωt), where the vector {right arrowover (f)} describes the spatial distribution of the excitation force.Similarly, the response of each measurement point is given by {rightarrow over (h)}(ω)e^(iωt). The components of {right arrow over (f)} and{right arrow over (h)} are complex because they contain phase as well asmagnitude information. It is this excitation and response data fromwhich modes shapes and natural frequencies may be extracted. However,for this data to be compatible with standard SISO modal analysismethods, it must preferably first be transformed to an appropriate modalanalysis coordinate system.

[0270] As indicated in the immediately preceding section, an appropriatecoordinate system that would allow this to occur is one in which thespatial distribution of the force, {right arrow over (f)}, is itself abasis vector. For simplicity, only the phase difference that occurs fromone blade to the next is included in the equation (57) below. In thecase of higher frequency applications, it may be necessary to alsoinclude the spatial variation of the force over the airfoil if more thanone family of modes interact. The spatial distribution of a travelingwave excitation has the form: $\begin{matrix}{{\overset{arrow}{f}}_{E} = {F_{\circ}\begin{bmatrix}^{0} \\^{{- }\quad {(\frac{2\quad \pi}{N})}E} \\\vdots \\^{{- }\quad {({N - 1})}{(\frac{2\quad \pi}{N})}E}\end{bmatrix}}} & (57)\end{matrix}$

[0271] where E is the engine order of the excitation. Therefore, acoordinate system whose basis vectors are the N possible values of{right arrow over (f)}, corresponding to all N distinct engine orderexcitations, 0 through N−1, may be used as a basis. The basis vectorsare complete and orthogonal.

[0272] The vectors {right arrow over (f)} and {right arrow over (h)} aretransformed into this modal analysis coordinate system by expressingthem as a sum of the basis vectors. Denoting the basis vectors as theset {{right arrow over (b)}₀, {right arrow over (b)}₁, . . . ,{rightarrow over (b)}_(N−1)}, this summation takes the form, $\begin{matrix}{\overset{arrow}{f} = {\sum\limits_{m = 0}^{N - 1}\quad {{\overset{\_}{f}}_{m}{\overset{arrow}{b}}_{m}}}} & ( {58a} ) \\{{\overset{arrow}{h}(\omega)} = {\sum\limits_{m = 0}^{N - 1}\quad {{\overset{\_}{h}(\omega)}_{m}{\overset{arrow}{b}}_{m}}}} & ( {58b} )\end{matrix}$

[0273] where the coefficients {overscore (f)}_(m) and {overscore(h)}(ω)_(m) describe the value of the m^(th) coordinate in the modalanalysis domain. To identify the values of these coefficients,orthogonality may be used. This is a general approach that may beapplicable for any orthogonal coordinate system. However, for the caseof traveling wave excitations, the coordinate transformation may besimplified.

[0274] Consider the n^(th) element of the vectors in equations (58). Forconvenience, let all vector indices run from 0 to N−1. Thus, theseelements may be expressed as, $\begin{matrix}{f_{n} = {\sum\limits_{m = 0}^{N - 1}\quad {{\overset{\_}{f}}_{m}^{{- }\quad {(\frac{2\quad \pi}{N})}m\quad n}}}} & ( {59a} ) \\{{h(\omega)}_{n} = {\sum\limits_{m = 0}^{N - 1}\quad {{\overset{\_}{h}(\omega)}_{m}^{{- }\quad {(\frac{2\quad \pi}{N})}m\quad n}}}} & ( {59b} )\end{matrix}$

[0275] where the exponential term is the n^(th) component of the basisvector {right arrow over (b)}_(m). Equation (59) is the inverse discreteFourier Transform (DFT⁻¹) of {overscore (f)}. This relation allows tostate the transformation between physical coordinates and the modalanalysis domain in the simpler form,

{right arrow over (f)}=DFT⁻¹{{overscore (f)}}  (60a)

{right arrow over (h)}=DFT⁻¹{{overscore (h)}}  (60b)

and conversely,

{overscore (f)}=DFT{{right arrow over (f)}}  (61a)

{overscore (h)}=DFT{{right arrow over (h)}}  (61b)

[0276] where DFT is the discrete Fourier Transform of the vector.

[0277] By applying equation (61), the force and response vectors may betransformed to the modal analysis coordinate system. Due to the presentselection of basis vectors, the resulting vector {overscore (f)} willcontain only one nonzero term that corresponds to the engine order ofthe excitation, i.e., a 5E excitation (fifth engine order excitation)will produce a nonzero term in element 5 of {overscore (f)}. Thisindicates that within the modal analysis domain, only the E^(th)coordinate has been excited. Therefore, {overscore (h)}(ω) representscolumn E of the FRF matrix.

[0278] The transformed response data, {overscore (h)}(ω), may now beanalyzed using standard SISO modal analysis algorithms. The resultingmodes will also be in the modal analysis coordinate system, and must beconverted back to physical coordinates though an inverse discreteFourier transform given, for example, in equation (60). These identified(mistuned) modes and natural frequencies may in turn be used as inputsto FMM ID methods to determine the mistuning of a bladed disk from itsresponse to an engine order excitation.

[0279] There are two further details of this method. First, for thepurpose of convenience of notation, the indices of all matrices andvectors are numbered from 0 to N−1. However, most modal analysissoftware packages use a numbering convention that starts at 1.Therefore, an E^(th) coordinate excitation in the present notationcorresponds to an (E+1)^(th) coordinate excitation in the standardconvention. This must be taken into account when specifying the“excitation point” in the modal analysis software. Second, thecoordinate transformation described herein is based on a set of complexbasis vectors. Because the modes are extracted in the modal analysisdomain they will be highly complex, even for lightly damped systems.Thus it may be necessary to use a modal analysis software package thatcan properly handle highly complex mode shapes. In one embodiment, theMODENT Suite by ICATS was used. Information about MODENT may be obtainedfrom Imregun, M., et al., 2002, MODENT2002, ICATS, London, UK,http://www.icats.co.uk.

[0280] 5.2 Experimental Test Cases

[0281] This section presents two experimental test cases of thetraveling wave system identification technique. In the first example, anintegrally bladed fan (IBR) was excited with a traveling wave while itwas in a stationary configuration. Because the IBR was stationary, itwas easier to make very accurate response measurements using a laservibrometer. Thus, this example may serve as a benchmark test of thetraveling wave identification theory. Then, in the second example, themethod's effectiveness on a rotor that is excited in a spin pit underrotating conditions is explored. The amplitude and phase of the responsewere measured using an NSMS system; NSMS is a non-contacting measurementmethod which is commonly used in the gas turbine industry for rotatingtests. The NSMS technology may be used with the traveling wave systemidentification technique to determine the IBR's mistuning from itsengine order response.

[0282] 5.2.1 Stationary Benchmark

[0283] An integrally bladed fan was tested using the traveling waveexcitation system at Wright Patterson Air Force Base's Turbine EngineFatigue Facility as discussed in Jones K. W., and Cross, C. J., 2003,“Traveling Wave Excitation System for Bladed Disks,” Journal ofPropulsion and Power, 19(1), pp. 135-141. Because the facility's testsystem used an array of phased electromagnets to generate a travelingwave excitation, the bladed disk remained stationary during the test.The experiment was performed with the fan placed on a rubber mat toapproximate a free boundary condition. First, the IBR was intentionallymistuned by fixing a different mass to the leading edge tip of eachblade with wax. The masses ranged between 0 and 7 g, and were selectedrandomly. Then, to obtain a benchmark measure of the mistuned fan's modeshapes, a standard SISO modal analysis test was performed. Specifically,a single electromagnet was used to excite one blade over the frequencyrange of the first bending modes while the response was measured at allsixteen (16) blades with a Scanning Laser Doppler Vibrometer (SLDV). Themodes were then extracted from the measured FRFs using the commerciallyavailable MODENT modal analysis package.

[0284] Next, to validate the traveling wave modal analysis method, thefan was excited using a 5^(th) engine order traveling wave excitation.Again, the response of each blade was measured using the SLDV. The bladeresponses to the traveling wave excitation were transformed usingequation (61) and then analyzed with MODENT to extract the transformedmodes. Because MODENT numbers its coordinates starting at 1 (0E), a 5Eexcitation corresponds to the excitation of coordinate 6. Therefore, inthe mode extraction process, it was specified that the excitation wasapplied at the 6^(th) coordinate. Lastly, equation (60) was used totransform the resulting modes back to physical coordinates.

[0285] The modes measured through the traveling wave test were thencompared with those from the benchmark analysis. FIGS. 40(a), (b) and(c) show a comparison of the representative mode shape extracted fromthe traveling wave response data with benchmark results. FIG. 40 thusshows several representative sets of mode shape comparisons that rangefrom nearly tuned-looking modes (e.g., FIG. 40(a)) to modes that arevery localized (e.g., FIG. 40(c)). In all cases in FIG. 40, the modesfrom the traveling wave and SISO benchmark methods agreed quite well. Inaddition, the natural frequencies were also accurately identified as canbe seen from FIG. 41, which depicts comparison of the naturalfrequencies extracted from the traveling wave response data with thebenchmark results. Thus, the traveling wave modal analysis method may beused to determine the modes and natural frequencies of a bladed diskbased on its response to a traveling wave excitation.

[0286] It is discussed below that the resulting modes and naturalfrequencies can be used with FMM ID methods to identify the mistuning inthe bladed disk. Because most of the mistuning in the stationarybenchmark fan was caused by the attached masses, to a large extent themistuning was known. Therefore, these mass values may be used as abenchmark with which to assess the accuracy of the FMM ID results.

[0287] Because the mass values are to be used as a benchmark, themistuning caused by the masses must be isolated from the inherentmistuning in the fan. Therefore, a standard SISO modal analysis wasfirst performed on the rotor fan with the masses removed, and theresulting modal data was used as input to FMM ID (e.g., advanced FMMID). This resulted in an assessment of the IBR's inherent mistuning,expressed as a percent change in each sector's frequency.

[0288] Next, an FMM ID analysis was performed of the modes andfrequencies extracted from the traveling wave response of the rotor withmass-mistuning. The resulting mistuning represented the total effect ofthe masses and the IBR's inherent mistuning. To isolate the mass effect,the rotor's nominal mistuning was subtracted. Again, the resultingmistuning was expressed as a percent change in each sector's frequency.

[0289] To compare these mistuning values with the actual masses placedon the blade tips, each sector frequency change may be first translatedinto its corresponding mass. A calibration curve to relate these twoquantities was generated through two independent methods. First, thecalibration was determined through a series of finite element analysesin which known mass elements were placed on the tip of a blade, and thefinite element model was used to directly calculate the effect of themass elements on the corresponding sector's frequency. It is noted thatin this method a single blade disk sector of the tuned bladed disk withcyclic symmetric boundary conditions applied to the disk was used.Further, changing the phase in the cyclic symmetric boundary conditionhad only a slight effect on the results (the results shown in FIG. 42corresponded to a phase constraint of 90 degrees). While this method wassufficient in this case, there are often times when a finite elementmodel is not available. For such cases, a similar calibration curve canbe generated experimentally by varying the mass on a single blade, andrepeating the FMM ID analysis. This experimental method was performed asan independent check of the calibration. Both approaches gave verysimilar results, as can be seen from the plot in FIG. 42, which shows acalibration curve relating the effect of a unit mass on a sector'sfrequency deviation in a stationary benchmark. For the range of massesused in this experiment, it was found that mass and sector frequencychange were linearly related as shown in FIG. 42. The calibration curveof FIG. 42 was then used to translate the identified sector frequencychanges into their corresponding masses.

[0290]FIG. 43 shows the comparison between the mass mistuning identifiedthrough traveling wave FMM ID with the values of the actual massesplaced on each blade tip (i.e., the input mistuning values). As can beseen from FIG. 43, the agreement between the mistuning obtained usingthe traveling wave system identification method and the benchmark valuesis quite good. Thus, by combining the traveling wave modal analysismethod with FMM ID, the mistuning in a bladed disk from its travelingwave response can be determined.

[0291] 5.2.2 Rotating Test Case

[0292] In the example in section 5.2.1, the traveling wave modalanalysis method was verified using a stationary benchmark rotor.However, if the method is to be applicable to conventional bladed disks,it may be desirable to make response measurements under rotatingconditions. This second test case assesses if the measurement techniquescommonly used in rotating tests are sufficiently accurate to be usedwith FMM ID to determine the mistuning in a bladed disk.

[0293] For this second case, another rotor fan was considered. To obtaina benchmark measure of the rotor's mistuning in its first bending modes,an impact hammer and a laser vibrometer were used to perform a SISOmodal analysis test. The resulting modes and natural frequencies werethen used as input to FMM ID to determine the fan's mistuning.

[0294] Next, the fan was tested in the spin pit facility at NASA GlennResearch Center. The NASA facility used an array of permanent magnets togenerate an eddy current excitation that drove the blades. The bladeresponse was then measured with an NSMS system. For this test, the fanwas driven with a 7E excitation, over a rotational speed range of 1550to 1850 RPM. The test was performed twice, at two different accelerationrates. The NSMS signals were then processed to obtain the amplitude andphase of each blade as a function of its excitation frequency. FIGS.44(a) and (b) show tracking plots of blade amplitudes as a function ofexcitation frequency for two different acceleration rates. The NSMSsystem measured the amplitude and phase of each blade once perrevolution. Thus, the data taken at the slower acceleration rate (FIG.44(b)) had a higher frequency resolution than that obtained from thefaster acceleration rate (FIG. 44(a)). However, in both cases, the datawas significantly noisier than the measurements obtained in the previousexample (in section 5.2.1) using an SLDV.

[0295] Next, the traveling wave system identification method was appliedto extract the mode shapes from the response data. First, themeasurements were transformed to the modal analysis domain by usingequation (61), and the mode shapes and natural frequencies wereextracted with MODENT. The extracted modes were then transformed back tothe physical domain through equation (60). Finally, the resulting modesand frequencies were used as input to FMM ID to identify the fan'smistuning.

[0296] The mistuning identified from the two spin pit tests was thencompared with the benchmark values. FIGS. 45(a) and (b) illustrate thecomparison of the mistuning determined through the traveling wave systemidentification method with benchmark values for two differentacceleration rates. In the case of the faster acceleration rate (FIG.45(a)), the trends of the mistuning pattern were identifiable, but themistuning values for all blades were not accurately determinable. Thereduced accuracy may be attributed to difficulty in extracting accuratemode shapes from data with such coarse frequency resolution. However,the frequency resolution of the data measured at a slower accelerationrate (FIG. 45(b)) was three times higher than the case for fasteracceleration. Thus, when FMM ID was applied to this higher resolutiondata set, the agreement between the traveling wave based ID and thebenchmark values was significantly improved, as can be seen from FIG.45(b). Thus, with adequate frequency resolution, NSMS measurements canbe used to determine the mistuning of a bladed disk under rotatingconditions.

[0297] Thus, NSMS measurements (from traveling wave excitation) may beused to elicit system mode shapes (blade number vs. displacement) andnatural frequencies. The modes and natural frequency data may then beinput to, for example, advanced FMM ID to infer frequency mistuning ofeach blade in a bladed disk and, thus, to predict the disk's forcedresponse.

[0298] There are a number of advantages to performing systemidentification based on a bladed disk's response to a traveling waveexcitation. First, it allows the use of data taken in a spin pit orstage test to determine a rotor's mistuning. In this way, the identifiedmistuning may include all effects present during the test conditions,i.e., centrifugal stiffening, gas loading, mounting conditions, as wellas temperature effects. The effect of centrifugal loading onconventional bladed disks may also be analyzed using FMM ID.

[0299] Although FMM ID theoretically only needs measurements of one ortwo modes, the method's robustness and accuracy may be greatly improvedwhen more modes are included. For certain bladed disks, a singletraveling wave excitation can be used to measure more modes than wouldbe possible from a single point excitation test. For example, in ahighly mistuned rotor that has a large number of localized modes, it maybe hard to excite all of these modes with only one single pointexcitation test, because the excitation source will likely be at a nodeof many of the modes. Therefore, to detect all of the mode shapes, thetest must be repeated at various excitation points. However, if thesystem is driven with a traveling wave excitation, all localized modescan generally be excited with just a single engine order excitation. Themore localized a mode becomes in physical coordinates, the more extendedit will be in the modal analysis coordinate system. Thus in highlymistuned systems, one engine order excitation can often provide moremodal information than several single point excitations.

[0300] The traveling wave system identification method may form thebasis of an engine health monitoring system. If a blade develops acrack, its frequency will decrease. Thus, by analyzing blade vibrationin the engine, the traveling wave system identification method coulddetect a cracked blade. A health monitoring system of this form may usesensors, such as NSMS, to measure the blade vibration. The measurementsmay be filtered to isolate an engine order response, and then analyzedusing the traveling wave system identification method to measure therotor's mistuning, which can be compared with previous measurements toidentify if any blade's frequency has changed significantly, thusidentifying potential cracks. It may be possible to develop a modeextraction method that does not require user interaction—i.e., anautomated modal analysis method which is tailored to a specific piece ofhardware.

[0301] The traveling wave system identification method discussedhereinabove may be extended to any structure subjected to a multi-pointexcitation in which the driving frequencies are consistent from oneexcitation point to the next. This allows structures to be tested in amanner that more accurately simulates their actual operating conditions.

[0302] [6] Conclusion

[0303]FIG. 46 illustrates an exemplary process flow depicting variousblade sector mistuning tools discussed herein. The flow chart in FIG. 46summarizes how the FMM and FMM ID methods discussed hereinbefore may beused to predict bladed disk system mistuning in stationary as well asrotating disks. For simplicity, the FMM discussion presentedhereinbefore addressed mistuning in mode families that are fairlyisolated in frequency (i.e., first two or three families of modes).Modeling mistuning in these modes may be relevant to the problem offlutter as discussed in Srinivasan, A. V., 1997, “Flutter and ResonantVibration Characteristics of Engine Blades,” Journal of Engineering forGas Turbines and Power, 119, 4, pp. 742-775. However, as mentionedbefore, the applicability of various FMM methodologies discussedhereinbefore may not be necessarily limited to an isolated family ofmodes.

[0304] Referring to FIG. 46, measurements (block 68) of the mode shapesand natural frequencies of a mistuned bladed disk (block 70) may beinput to various FMM ID methods (block 72) to infer the mistuning ineach blade/disk sector. The advanced FMM ID method can also calculatethe natural frequencies that the system would have if it were tuned,i.e., was perfectly periodic. The detailed discussion of blocks 68, 70,and 72 has been provided under parts 1 through 3 hereinabove. Becausemode shapes measurements are generally made on stationary systems, theresulting mistuning often corresponds to a non-rotating bladed disk.However, centrifugal forces that are present while the disk rotates canalter the mistuning. Thus, mistuning extrapolation (block 74) may beperformed to correct the mistuning from a stationary rotor for theeffects of centrifugal stiffening. Mistuning extrapolation has beendiscussed under part-4 hereinabove. The FMM methodology (including FMMID methods) may be coupled with a modal summation algorithm to calculatethe forced response (block 76) of a bladed disk based on the mistuningidentified in the previous steps. The discussion of forced responseanalysis (block 76) has been provided hereinabove at various locationsunder parts 1 through 3. Further, as discussed under part-5 above, themode shapes and natural frequencies of a bladed disk may be extractedfrom its response to a traveling wave excitation (blocks 78, 80). Thus,by combining the traveling wave modal analysis technique (block 80)(which may use NSMS measurements identified at block 78) with the FMM IDsystem identification methods, mistuning in a bladed disk can bedetermined while the disk is under actual operating conditions.

[0305] The vibratory response of a turbine engine bladed disk is verysensitive to mistuning. As a result, mistuning increases its resonantstress and contributes to high cycle fatigue. The vibratory response ofa mistuned bladed disk system may be predicted by the FundamentalMistuning Model (FMM) because of its identification of parameters—thetuned system frequencies and the sector frequency deviations—thatcontrol the mode shapes and natural frequencies of a mistuned bladeddisk. Neither the geometry of the system nor the physical cause of themistuning may be needed. Thus, FMM requires little or no interactionwith finite element analysis and is, thus, extremely simple to apply.The simplicity of FMM provides an approach for making bladed disks lesssensitive to mistuning—at least in isolated families of modes. Of thetwo parameters that control the mistuned modes of the system, one is themistuning itself which has a standard deviation that is typically knownfrom past experience. The only other parameters that affect the mistunedmodes are the natural frequencies of the tuned system. Consequently, thesensitivity of the system to mistuning can be changed only to the extentthat physical changes in the bladed disk geometry affect thesefrequencies. For example, if the disk were designed to be more flexible,then the frequencies of the tuned system would be spread over a broaderrange, and this may reduce the system's sensitivity to mistuning.

[0306] The FMM ID methods use measurements of the mistuned system as awhole to infer its mistuning. The measurements of the system mode shapesand natural frequencies can be obtained in laboratory test throughstandard modal analysis procedures. The high sensitivity of system modesto small variations in mistuning causes measurements of those modesthemselves to be an accurate basis for mistuning identification. BecauseFMM ID does not require individual blade measurements, it isparticularly suited to integrally bladed rotors. The basic FMM ID methodrequires the natural frequencies of the tuned system as an input. Themethod is useful for comparing the change in a components mistuning overtime, because each calculation will be based about a consistent set oftuned frequencies. The advanced FMM ID method, on the other hand, doesnot require any analytical data. The approach is completely experimentaland determines both the mistuning and the tuned system frequencies ofthe rotor.

[0307] Effects of centrifugal stiffening on mistuning may be identifiedon a stationary IBR using FMM ID and FMM, and extrapolated to engineoperating conditions to predict the system's forced response at speed.Further, in conventional bladed disks, centrifugal forces may causechanges in the contact conditions at the blade/disk attachment tosubstantially alter the system's mistuning. This behavior may not beaccounted for in the mistuning extrapolation method. In that case, themode shapes and natural frequencies of a rotating bladed disk may beextracted from measurements of its forced response (e.g., traveling waveexcitation) and the results may then be combined with FMM ID todetermine the mistuning present at operating conditions.

[0308] It is observed that, besides centrifugal loading, other factorsmay also be present in the engine that can affect its mistuned response.These may include: temperature effects, gas bending stresses, how thedisk is constrained in the engine, and how the teeth in the attachmentchange their contact if the blades are conventionally attached to thedisk. Except for the constraints on the disk, these additional effectsmay be relatively unimportant in integrally bladed compressor stages.The disk constraints can be taken into account by performing the systemidentification (using, for example, an FMM ID method) on the IBR afterthe full rotor is assembled. Thus, the FMM ID methodology presentedherein may be used to predict the vibratory response of actualcompressor stages so as to determine which blades may be instrumented,interpret test data, and relate the vibratory response measured in theCRF to the vibration that will occur in the fleet as a whole.

[0309] The traveling wave modal analysis method discussed hereinbeforemay detect the presence of a crack in an engine blade by analyzing bladevibrations because a crack will decrease a blade's frequency ofvibration. This method, thus, may be used with on-board sensors tomeasure blade response during engine accelerations. The measurements maybe filtered to isolate an engine order response, and then analyzed usingthe traveling wave system identification method. The identifiedmistuning may then be compared with previous results to determine if anyblade's frequency has changed significantly, thus identifying potentialcracks. The FMM and FMM ID methods may be applied to regions of highermodal density using Subset of Nominal Modes (SNM) method.

[0310] The foregoing describes development of a reduced order modelcalled the Fundamental Mistuning Model (FMM) to accurately predictvibratory response of a bladed disk system. FMM may describe the normalmodes and natural frequencies of a mistuned bladed disk using only itstuned system frequencies and the frequency mistuning of each blade/disksector (i.e., the sector frequencies). If the modal damping and theorder of the engine excitation are known, then FMM can be used tocalculate how much the vibratory response of the bladed disk willincrease because of mistuning when it is in use. The tuned systemfrequencies are the frequencies that each blade-disk and blade wouldhave were they manufactured exactly the same as the nominal designspecified in the engineering drawings. The sector frequenciesdistinguish blade-disks with high vibratory response from those with alow response. The FMM identification methods—basic and advanced FMM IDmethods-use the normal (i.e., mistuned) modes and natural frequencies ofthe mistuned bladed disk measured in the laboratory to determine sectorfrequencies as well as tuned system frequencies. Thus, one use of theFMM methodology is to: identify the mistuning when the bladed disk is atrest, to extrapolate the mistuning to engine operating conditions, andto predict how much the bladed disk will vibrate under the operating(rotating) conditions.

[0311] In one embodiment, the normal modes and natural frequencies ofthe mistuned bladed disk are directly determined from the disk'svibratory response to a traveling wave excitation in the engine. Thesemodes and natural frequency may then be input to the FMM ID methodologyto monitor the sector frequencies when the bladed disk is actuallyrotating in the engine. The frequency of a disk sector may change if theblade's geometry changes because of cracking, erosion, or impact with aforeign object (e.g., a bird). Thus, field calibration and testing ofthe blades (e.g., to assess damage from vibrations in the engine) may beperformed using traveling wave analysis and FMM ID methods together.

[0312] It is noted that because the FMM model can be generatedcompletely from experimental data (e.g., using the advanced FMM IDmethod), the tuned system frequencies from advanced FMM ID may be usedto validate the tuned system finite element model used by industry.Further, FMM and FMM ID methods are simple, i.e., no finite element massor stiffness matrices are required. Consequently, no special interfacesare required for FMM to be compatible with a finite element model.

[0313] While the disclosure has been described in detail and withreference to specific embodiments thereof, it will be apparent to oneskilled in the art that various changes and modifications can be madetherein without departing from the spirit and scope of the embodiments.Thus, it is intended that the present disclosure cover the modificationsand variations of this disclosure provided they come within the scope ofthe appended claims and their equivalents.

What is claimed is:
 1. A method, comprising: obtaining a set ofvibration measurements that provides frequency deviation of each bladeof a bladed disk system from the tuned frequency value of said blade andnominal frequencies of said bladed disk system when tuned; andcalculating the mistuned modes and natural frequencies of said bladeddisk system from said blade frequency deviations and said nominalfrequencies.
 2. The method of claim 1 wherein each said blade includes acorresponding blade-disk sector in said disk system.
 3. The method ofclaim 1, wherein said obtaining includes receiving said set of vibrationmeasurements for an isolated family of modes of said disk system.
 4. Themethod of claim 1, further comprising: further obtaining a set of valuesfor modal damping and engine order of excitation for said bladed disksystem; and further calculating amplitude magnification due to mistuningfor at least one blade in said bladed disk system using values for saidmodal damping and said engine order of excitation.
 5. The method ofclaim 2, wherein said calculating includes solving: (Ω°²+2Ω°{overscore(Ω)}Ω°){right arrow over (β)}_(j)=ω_(j) ²{right arrow over (β)}_(j)where Ω° is a diagonal matrix of the nominal frequencies of said bladeddisk system when tuned; {overscore (Ω)} is a matrix containing thediscrete Fourier transforms of the blade frequency deviations; {rightarrow over (β)}_(j) is a vector containing weighting factors thatdescribe the j^(th) mistuned mode as a sum of tuned modes; and ω_(j) isthe natural frequency of the j^(th) mistuned mode.
 6. The method ofclaim 2, wherein physical displacements of the n^(th) blade in thej^(th) mistuned mode are proportional to:$x_{n} = {\sum\limits_{m = 0}^{N - 1}\quad {\beta_{jm}^{{imn}\frac{2\pi}{N}}}}$

where x_(n) represents physical displacements of the n^(th) blade;β_(jm) is a weighting factor for the j^(th) mistuned mode with m=0, 1, .. . , N−1.
 7. The method of claim 1, wherein said obtaining includesgenerating said nominal frequencies of said bladed disk system whentuned using a finite element analysis.
 8. The method of claim 1, whereinsaid obtaining includes importing calculated values to form at least apart of said set of vibration measurements.
 9. The method of claim 1,wherein said obtaining includes exciting a plurality of blades in saidbladed disk system and measuring the vibration of each of said pluralityof blades.
 10. The method of claim 1, wherein said obtaining includesrandomly generating at least a part of said set of vibrationmeasurements.
 11. The method of claim 10, wherein said randomlygenerating includes randomly generating at least said part of said setof vibration measurements using a Monte Carlo simulation.
 12. A method,comprising: obtaining nominal frequencies of a bladed disk system whentuned; measuring at least one mistuned mode and natural frequency ofsaid bladed disk system; and calculating mistuning of at least one bladein said bladed disk system from only said nominal frequencies and saidat least one mistuned mode and natural frequency.
 13. The method ofclaim 12, wherein said obtaining includes obtaining said nominalfrequencies for an isolated family of modes of said bladed disk systemwhen tuned.
 14. The method of claim 12, wherein said obtaining includesobtaining said nominal frequencies using a finite element analysis. 15.The method of claim 14, wherein said finite element analysis includesfinite element analysis of a tuned, cyclic symmetric model of a singleblade-disk sector in said bladed disk system.
 16. The method of claim12, wherein said measuring includes measuring said at least one mistunedmode and natural frequency using a modal analysis.
 17. The method ofclaim 16, wherein said measuring using said modal analysis includes:measuring a set of frequency response functions of said bladed disksystem; and extracting said at least one mistuned mode and naturalfrequency from said frequency response functions.
 18. The method ofclaim 17, wherein said extracting includes extracting said at least onemistuned mode and natural frequency using modal curve fitting.
 19. Themethod of claim 12, wherein said blade includes a correspondingblade-disk sector in said bladed disk system.
 20. The method of claim19, wherein said calculating includes solving: ${\begin{bmatrix}{2\Omega^{{^\circ}}\Gamma_{1}} \\{2\Omega^{{^\circ}}\Gamma_{2}} \\\vdots \\{2\Omega^{{^\circ}}\Gamma_{m}}\end{bmatrix}\overset{\overset{arrow}{\_}}{\omega}} = \begin{bmatrix}{( {{\omega_{1}^{2}I} - \Omega^{{^\circ}2}} ){\overset{}{\beta}}_{1}} \\{( {{\omega_{2}^{2}I} - \Omega^{{^\circ}2}} ){\overset{}{\beta}}_{2}} \\\vdots \\{( {{\omega_{m}^{2}I} - \Omega^{{^\circ}2}} ){\overset{}{\beta}}_{m}}\end{bmatrix}$

where Ω° is a diagonal matrix of the nominal frequencies of said bladeddisk system when tuned; Γ_(j) is a matrix composed from the elements inthe vector {right arrow over (γ)}_(j) where {right arrow over(γ)}=Ω°{right arrow over (β)}_(j) {right arrow over (β)}_(j) is a vectorcontaining weighting factors that describe the j^(th) mistuned mode as asum of tuned modes; I is the identity matrix; {overscore (ω)} is avector of mistuning parameters; and ω_(j) is the natural frequency ofthe j^(th) mistuned mode.
 21. The method of claim 20, further comprisingusing a least squares fit method to determine the vector {overscore (ω)}that best fits all the measured data.
 22. The method of claim 20,wherein the vector {overscore (ω)} is related to a physical sectormistuning by the equation${\Delta\omega}_{\psi}^{(s)} = {\sum\limits_{p = 0}^{N - 1}\quad {^{{{- }\quad s\quad p\frac{2\pi}{N}},}{\overset{\_}{\omega}}_{p}}}$

where Δω_(ψ) ^((s)) is the sector frequency deviation of the s^(th)blade-disk sector; and {overscore (ω)}_(p) is p^(th) mistuning parameterin the vector {overscore (ω)}.
 23. A method, comprising: measuring a setof mistuned modes and natural frequencies of a bladed disk system; andcalculating mistuning of at least one blade in said bladed disk systemand nominal frequencies of said bladed disk system when tuned by usingonly said set of mistuned modes and natural frequencies.
 24. The methodof claim 23, further comprising validating a finite element model ofsaid bladed disk system using said nominal frequencies of the tunedbladed disk system.
 25. The method of claim 23, wherein calculating saidnominal frequencies includes calculating said nominal frequencies usinga finite element model of said bladed disk system treating each blade insaid bladed disk system as identical and also using said set of mistunedmodes and natural frequencies.
 26. The method of claim 23, wherein saidcalculating includes calculating said nominal frequencies for anisolated family of modes of said bladed disk system when tuned.
 27. Themethod of claim 23, wherein said calculating is performed iteratively.28. The method of claim 23, wherein said blade includes a correspondingblade-disk sector in said bladed disk system.
 29. The method of claim28, wherein a mean value of mistuning of at least one blade-disk sectoris zero.
 30. The method of claim 29, wherein said calculating includessolving: ${\begin{bmatrix}\overset{\sim}{B} & {2( \overset{\sim}{\Omega^{{^\circ}}\Gamma} )} \\0 & \overset{}{c}\end{bmatrix}\begin{bmatrix}{\overset{}{\lambda}}^{{^\circ}} \\\overset{\overset{arrow}{\_}}{\omega}\end{bmatrix}} = \begin{bmatrix}\overset{\overset{\sim}{}}{r} \\0\end{bmatrix}$

where {tilde over (B)} is a stacked matrix composed from the elements of{right arrow over (β)}_(j), which is a vector containing weightingfactors that describe the j^(th) mistuned mode as a sum of tuned modes;({tilde over (Ω°Γ)}) is a stacked matrix of Ω°Γ_(j), where Ω° is adiagonal matrix of the nominal frequencies of said bladed disk systemwhen tuned and Γ_(j) is a matrix composed from the elements in thevector {right arrow over (γ)}_(j) where {right arrow over(γ)}_(j)=Ω°{right arrow over (β)}_(j) {right arrow over (c)} is a rowvector whose first element is 1 and whose remaining elements are zero;{overscore (ω)} is a vector of mistuning parameters; {overscore (λ)}° isa vector of the tuned frequencies squared; and {right arrow over ({tildeover (r)})} is the vector given by the following $\quad\begin{bmatrix}{\omega_{1}^{2}{\overset{}{\beta}}_{1}} \\{\omega_{2}^{2}{\overset{}{\beta}}_{2}} \\\vdots \\{\omega_{M}^{2}{\overset{}{\beta}}_{M}}\end{bmatrix}$

where ω_(j) is the natural frequency of the j^(th) mistuned mode. 31.The method of claim 30, wherein the vector {overscore (ω)} is related toa physical sector mistuning by the equation${\Delta \quad \omega_{\psi}^{(s)}} = {\sum\limits_{p = 0}^{N - 1}\quad {^{{- }\quad {sp}\frac{2\pi}{N}}{\overset{\_}{\omega}}_{p}}}$

where Δω_(ψ) ^((s)) is the sector frequency deviation of the s^(th)blade-disk sector; and {overscore (ω)}_(p) is a p^(th) mistuningparameter in the vector {overscore (ω)}.
 32. A method, comprising:identifying mistuning in a bladed disk when said disk is not rotating;extrapolating said mistuning to apply to a condition when said bladeddisk is rotating; and predicting from said extrapolation vibratoryresponse of said bladed disk under operating conditions.
 33. A method,comprising: obtaining frequency response data of each blade in a bladeddisk system to a traveling wave excitation; transforming data related tospatial distribution of said traveling wave excitation and saidfrequency response data; and determining a set of mistuned modes andnatural frequencies of said bladed disk system using data obtained fromsaid transformation.
 34. The method of claim 33, wherein said travelingwave excitation has a spatially-invariant frequency profile.
 35. Themethod of claim 33, wherein said obtaining includes measuring amplitudeand phase of displacement of each said blade as a function of saidtraveling wave excitation.
 36. The method of claim 33, wherein saiddetermining includes converting said data obtained from saidtransformation into a set of physical coordinates.
 37. The method ofclaim 33, further comprising: calculating mistuning of a blade in saidbladed disk system and nominal frequency of said bladed disk system whentuned by using said set of mistuned modes and natural frequencies. 38.The method of claim 37, further comprising validating a finite elementmodel of said bladed disk system using said nominal frequencies of thetuned bladed disk system.
 39. The method of claim 37, whereincalculating said nominal frequencies includes calculating said nominalfrequencies using a finite element model of said bladed disk systemtreating each blade in said bladed disk system as identical and alsousing said set of mistuned modes and natural frequencies.
 40. The methodof claim 37, wherein said calculating includes calculating said nominalfrequencies for an isolated family of modes of said bladed disk systemwhen tuned.
 41. The method of claim 37, wherein said calculating isperformed iteratively.
 42. The method of claim 37, wherein said bladeincludes a corresponding blade-disk sector in said bladed disk system.43. The method of claim 42, wherein a mean value of mistuning of atleast one blade-disk sector is zero.
 44. The method of claim 43, whereinsaid calculating includes solving: ${\begin{bmatrix}\overset{\sim}{B} & {2( \overset{\sim}{\Omega^{{^\circ}}\Gamma} )} \\0 & \overset{}{c}\end{bmatrix}\begin{bmatrix}{\overset{}{\lambda}}^{{^\circ}} \\\overset{\overset{arrow}{\_}}{\omega}\end{bmatrix}} = \begin{bmatrix}\overset{\overset{\sim}{}}{r} \\0\end{bmatrix}$

where {tilde over (B)} is a stacked matrix composed from the elements of{right arrow over (β)}_(j), which is a vector containing weightingfactors that describe the j^(th) mistuned mode as a sum of tuned modes;({tilde over (Ω°Γ)}) is a stacked matrix of Ω°Γ_(j), where Ω° is adiagonal matrix of the nominal frequencies of said bladed disk systemwhen tuned and Γ_(j) is a matrix composed from the elements in thevector {right arrow over (γ)}_(j) where {right arrow over(γ)}_(j)=Ω°{right arrow over (β)}_(j) {right arrow over (c)} is a rowvector whose first element is 1 and whose remaining elements are zero;{overscore (ω)} is a vector of mistuning parameters; {right arrow over(λ)}° is a vector of the tuned frequencies squared; and {right arrowover ({tilde over (r)})} is the vector given by the following$\quad\begin{bmatrix}{( {{\omega_{1}^{2}I} - \Omega^{{^\circ}2}} ){\overset{}{\beta}}_{1}} \\{( {{\omega_{1}^{2}I} - \Omega^{{^\circ}2}} ){\overset{}{\beta}}_{2}} \\\vdots \\{( {{\omega_{1}^{2}I} - \Omega^{{^\circ}2}} ){\overset{}{\beta}}_{m}}\end{bmatrix}$

where I is the identity matrix, and ω₁ is the natural frequency of the1^(st) mistuned mode.
 45. The method of claim 44, wherein the vector{overscore (ω)} is related to a physical sector mistuning by theequation${\Delta \quad \omega_{\psi}^{(s)}} = {\sum\limits_{p = 0}^{N - 1}\quad {^{{- }\quad {sp}\frac{2\pi}{N}}{\overset{\_}{\omega}}_{p}}}$

where Δω_(ψ) ^((s)) is the sector frequency deviation of the s^(th)blade-disk sector; and {overscore (ω)}_(p) is a pth mistuning parameterin the vector {overscore (ω)}.
 46. The method of claim 33, furthercomprising: obtaining nominal frequencies of said bladed disk systemwhen tuned; and calculating mistuning of a blade in said bladed disksystem from said nominal frequencies and said mistuned modes and naturalfrequencies.
 47. The method of claim 46, wherein said obtaining includesobtaining said nominal frequencies for an isolated family of modes ofsaid bladed disk system when tuned.
 48. The method of claim 46, whereinsaid obtaining includes obtaining said nominal frequencies using afinite element analysis.
 49. The method of claim 48, wherein said finiteelement analysis includes finite element analysis of a tuned, cyclicsymmetric model of a single blade-disk sector in said bladed disksystem.
 50. The method of claim 46, wherein said blade includes acorresponding blade-disk sector in said bladed disk system.
 51. Themethod of claim 33, wherein said obtaining includes: rotating saidbladed disk system; and exciting said rotating bladed disk system withpressure fluctuations.
 52. The method of claim 33, wherein said bladeincludes a corresponding blade-disk sector in said bladed disk system.53. The method of claim 33, wherein said transforming is performed froma physical coordinates domain to a modal analysis domain.
 54. The methodof claim 53, wherein said transforming is performed according to thefollowing equations: {overscore (f)}=DFT{{right arrow over(f)}}{overscore (h)}=DFT{{right arrow over (h)}} where {right arrow over(f)} is a vector that describes the spatial distribution of saidtraveling wave excitation; {right arrow over (h)} is a vector thatdescribes the frequency response of each measurement point to saidtraveling wave excitation; {overscore (f)} is a vector that is adiscrete Fourier Transform of the force vector {right arrow over (f)};and {overscore (h)} is a vector that is a discrete Fourier Transform ofthe response vector {right arrow over (h)}.
 55. A computer-readable datastorage medium containing a program code, which, when executed by aprocessor, causes said processor to perform the following: receive a setof vibration measurements that provides frequency deviation of eachblade of a bladed disk system from the tuned frequency value of saidblade and nominal frequencies of said bladed disk system when tuned; andcalculate the mistuned modes and natural frequencies of at least oneblade in said bladed disk system from said blade frequency deviationsand said nominal frequencies.
 56. A computer-readable data storagemedium containing a program code, which, when executed by a processor,causes said processor to perform the following: receive nominalfrequencies of a bladed disk system when tuned; further receive themistuned modes and natural frequencies of said bladed disk system; andcalculate mistuning of a blade in said bladed disk system only from saidnominal frequencies and said mistuned modes and natural frequencies. 57.A computer-readable data storage medium containing a program code,which, when executed by a processor, causes said processor to performthe following: receive a set of mistuned modes and natural frequenciesof a bladed disk system; and calculate mistuning of a blade in saidbladed disk system by using only said set of mistuned modes and naturalfrequencies.
 58. The data storage medium of claim 57, wherein saidprogram code, upon execution by said processor, causes said processor tofurther perform the following: calculate nominal frequencies of saidbladed disk system when tuned by using only said set of mistuned modesand natural frequencies.
 59. A computer-readable data storage mediumcontaining a program code, which, when executed by a processor, causessaid processor to perform the following: identify mistuning in a bladeddisk when said disk is not rotating; extrapolate said mistuning to applyto a condition when said bladed disk is rotating; and predict from saidextrapolation vibratory response of said bladed disk under operatingconditions.
 60. A computer-readable data storage medium containing aprogram code, which, when executed by a processor, causes said processorto perform the following: receive frequency response data of each bladein a bladed disk system to a traveling wave excitation; transform datarelated to spatial distribution of said traveling wave excitation andsaid frequency response data; and determine a set of mistuned modes andnatural frequencies of said bladed disk system using data obtained fromsaid transformation.